# Theory ofB(X)-Module: Algebraic Module Structure of Generally Unbounded Infinitesimal Generators

Authors: Yoritaka Iwata
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally unbounded infinitesimal generators. In conclusion, the concept of module over a Banach algebra is proposed as the generalization of the Banach algebra. As an application to mathematical physics, the rigorous formulation of a rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.


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## Body

## 1. Introduction

Based on the logarithmic representation of infinitesimal generators, a module over a Banach algebra is introduced. Let us call such an algebraic subject theBX-module, where X and BX stand for a Banach space and its operator algebra, respectively. The BX-module does not correspond only to the extension of the Banach algebra but also to the general authorization of the Lie algebra consisting of differential operators. This algebraic entity is an operator algebra being introduced based on the framework of logarithmic representation of operators. There are two concepts, which are to be bridged in this paper: a set of infinitesimal generators generating groups or semigroups of operators and the elements of the Lie algebra. The following statements are valid:
(i)
The sum of two closed operators are not necessarily a closed operator so that the sum of two infinitesimal generators are not necessarily an infinitesimal generator(ii)
The sum of two elements in the Lie algebra are necessarily an element of the Lie algebraHere is a contradiction in some general situations, as seen in the relation between the Lie group and the Lie algebra in which the Lie algebra corresponds to a set of infinitesimal generators. Besides, these two statements are true if the two infinitesimal generators are bounded operators. More substantially, the product cannot be justified without limiting ourselves to (sub)sets of bounded operators. In this paper, by means of the logarithmic representation of infinitesimal generators of invertible evolution operators, a set of generally unbounded infinitesimal generators is characterized as an algebraic module over a Banach algebra. The logarithm of operators is a key to make a bridge for these contradicting statements.The logarithm of an injective sectorial operator was introduced by Nollau [1] in 1969. After a long time, the logarithm of sectorial operators was studied again from 1990s [2–4], and its utility was established with respect to the definition of the logarithms of operators [5, 6] (for a review of sectorial operators, see Hasse [7]). While the sectorial operator has been a generic framework to define the logarithm of operators, the sectorial property is not generally satisfied by the evolution operators. In this sense, it is necessary to introduce a reasonable framework for defining the logarithm of nonsectorial infinitesimal generators.In this paper, the theory ofBX nor-module is introduced. The utility of the theory is confirmed in the application to the solvability of abstract Cauchy problem, the generalization of the Cole-Hopf transform, and the foundation of the rotation group in the latter parts of Sections 2, 3, and 4.This paper is the completion of the recent studies shown in Refs. [8–15]. By organizing the preceding works into a logical order, the several statements are renewed. First, the generalized version of logarithmic representation (Corollary 8) is possible using the concept of alternative infinitesimal generator. Even without any additional assumptions, it enables us to generalize the logarithmic representation for infinitesimal generators of noninvertible evolution operators. Although this fact is taken for granted in the lately published papers of Refs. [8–15], it is mentioned within a logical process for the first time. Second, although the relativistic formulation is introduced for changing the evolution direction as seen in the application of the Cole-Hopf transform, it should not be restricted to the application of the Cole-Hopf transform. The relativistic formulation of abstract evolution equation is a kind of generalization of abstract evolution equations. More clearly, it generalizes the concept of abstract evolution equation to the abstract equation. Consequently, the theory of BX-module is written in the relativistic form.


## 2. Logarithmic Representation of Operators

### 2.1. Banach Algebra

LetA,⋅ be a Banach space (for a textbook, see [16]). A mapping
(1)A×A⟶A,x,y⟶x⋅y,is called a multiplication on A, if it is bilinear and associative. ⋅ is said to be a submultiplicative norm if x⋅y≤xy for each x,y∈A. The Banach space A together with a multiplication and submultiplicative norm is called a Banach algebra.LetX be a Banach space. Denote by BX, the set of all bounded linear operators f:X⟶X. Then, A=BX is an example of a Banach algebra with multiplication as composition and norm defined by
(2)f=supfx:x≤1,where f∈BX. BX is called the operator algebra of X. Let A be a Banach algebra and let X be a Banach space. X is said to be
(i)
A left BanachA-module if there exists a bilinear mapping ⋅:A×X⟶X, a,x⟶a⋅x, called left module action, such that a⋅x≤ax and
(3)ab⋅x=a⋅b⋅x,a,b∈Aandx∈X.(ii)
A right BanachA-module if there exists a bilinear mapping ⋅:A×X⟶X, x,a⟶x⋅a, called right module action, such that x⋅a≤ax and
(4)x⋅ab=x⋅a⋅b,a,b∈Aandx∈X.(iii)
A BanachA-module if it is a left and right Banach A-module and
(5)a⋅x⋅b=a⋅x⋅b,a,b∈Aandx∈X.As an example of the Banach algebra,BX is taken in the following. Then, X is a Banach BX-module under
(6)BX×X⟶X,f,x⟶fx,X×BX⟶X,x,f⟶fx.The Banach space, Banach algebra, and BanachBX-module (BX-module, for short) are the basic concepts in this paper.


### 2.2. Two Parameter Group on Banach Spaces

All the discussion begins with the definition of groups on the Banach spaces that will be generalized to a well-defined semigroup in later sections. LetX,⋅ be a Banach space and BX its Banach algebra. In particular, BX is an example of a Banach algebra.A two parameter group is defined onX. Let X be a Banach space and T∈0,∞. A two parameter group on X is an operator valued mapping t,s⟶Ut,s from −T,T into BX with the semigroup properties:
(7)Ut,rUr,s=Ut,s,r,s,t∈−T,T,(8)Us,s=I,s∈−T,T,and the strong continuity; for eachs∈−T,T and x∈X, the map t⟶Ut,sx is continuous on s,T. Both Ut,s and Us,t are assumed to be well-defined to satisfy
(9)Us,tUt,s=Us,s=I,where Us,t corresponds to the inverse operator of Ut,s. Since Ut,sUs,t=Ut,t=I is also true, the commutation between Ut,s and Us,t follows. Operator Ut,s, which is called the evolution operator in the following, is a generalization of exponential function; indeed, the properties shown in equations (7)–(9) are satisfied by taking Ut,s as et−s. Evolution operator is an abstract concept of exponential function valid for both finite and infinite dimensional Banach spaces. Due to the validity of equation (9), the invertible evolution family is to be associated with some linear evolution equations of hyperbolic type and those of dispersive type. In the same context, the obtained results can be directly applied to some semilinear evolution equations (for a text book, see [17]). For example, the solutions of linear and nonlinear wave equations are written by the evolution operator Ut,s defined above.LetY be a dense Banach subspace of the Banach space X and the topology of Y be stronger than that of X. The space Y⊂X is assumed to be Ut,s-invariant; for any t,s satisfying −T≤t,s≤T,Ut,sY=Y. Following the definition of C0-(semi) group (cf. the assumption H2 in Section 5.3 of Pazy [18] or corresponding discussion in Kato [19, 20]), Ut,s trivially satisfy the boundedness in the present setting; there exist real numbers M and β such that
(10)Ut,sBX≤Meβt,Ut,sBY≤Meβt. that are practically reduced to
(11)Ut,sBX≤MeβT,Ut,sBY≤MeβT,when the interval is restricted to be finite −T,T. Since C0 semigroup theory [21–23], is essentially based on the Laplace transform of operators, the satisfaction of equation (10) is discussed here; Meβt in equation (10) arises from the condition for the existence theorem for the Laplace transforms (for example, see [24]), and M′=MeβT is regarded as a finite real number in the present setting.Next, for the well-definedUt,s, the counterpart of the logarithm in the abstract framework is introduced. There are two concepts associated with the logarithm of operators; one is the infinitesimal generator, and the other is t-differential of Ut,s. These two concepts are connected as follows.Definition 1 (preinfinitesimal generator).
For−T≤t,s≤T, the weak limit
(12)wlimh→0h−1Ut+h,s−Ut,sus=wlimh→0h−1Ut+h,t−IUt,sus,is assumed to exist for certain us, which is an element of a dense subspace Y of X. A linear operator At:Y⟶X is defined by
(13)Atut≔wlimh→0h−1Ut+h,t−Iut,for ut∈Y and −T≤t,s≤T. The operator At for a whole family Ut,s−T≤t,s≤T is called the preinfinitesimal generator.Lett-differential of Ut,s in a weak sense [13] be denoted by
(14)∂tUt,sus=AtUt,sus.Equation (14) is regarded as a differential equation satisfied by ut=Ut,sus that implies a relation between At and the logarithm:
(15)At=∂tUt,sUs,t.The relation betweenAt and the logarithm is discussed in the next Section 2.3. Preinfinitesimal generators are not necessarily infinitesimal generators without assuming a dense property of domain space Y in X. For example, in t-independent cases, an operator At defined by equation (13) is not necessarily a densely defined and closed linear operator, while At must be a densely defined and closed linear operator with its resolvent set included in λ∈ℂ:Reλ>β for At to be the infinitesimal generator. On the other hand, infinitesimal generators are necessarily preinfinitesimal generators. That is, only the exponentiability with a certain ideal domain is valid to the preinfinitesimal generators. The definition of preinfinitesimal generator is useful in terms of providing the algebraic structure. Let a set of preinfinitesimal generators be denoted by GX. It is trivial that BX⊂GX.


### 2.3. Logarithmic Representation of Preinfinitesimal Generator

The logarithmic representation of infinitesimal generator is introduced in order to clarify the structure of infinitesimal generators [8]. The logarithm of Ut,s is defined by the Dunford-Riesz integral [25]. The boundedness of Ut,s on X makes the problem rather easy. Indeed, the boundedness allows us to introduce the translation on the complex plane as a tool to realize the parallel displacement of the entire spectral set. On the other hand, two difficulties inherent to the logarithm
(i)
Singularity of logarithm at the origin(ii)
Multivalued property of the logarithmarise. By introducing a constant κ∈ℂ, the singularity can be handled. This simple treatment is definitely practical to well-define the logarithm of nonsectorial operators. By introducing a principal branch (denoted by “Log”) of the logarithm (denoted by “log”), the multivalued property is handled. Indeed, for any complex number z∈C, a branch of logarithm is defined by
(16)Logz=logz+iargZ,where Z is a complex number chosen to satisfy ∣Z∣=∣z∣, −π<argZ≤π, and argZ=argz+2nπ for a certain integer n.Lemma 2.
(logarithmic representation of operators). Lett and s satisfy 0≤t,s≤T. For a given Ut,s defined in Section 2.2, its logarithm is well defined; there exists a certain complex number κ satisfying
(17)LogUt,s+κI=12πi∫ΓLogλλI−Ut,s−κI−1dλ,where an integral path Γ, which excludes the origin, is a circle in the resolvent set of Ut,s+κI. Here, Γ is independent of t and s. LogUt,s+κI is bounded on X.
Proof. The logarithm Log holds the singularity at the origin so that it is necessary to show a possibility of taking a simple closed curve (integral path) excluding the origin in order to define the logarithm by means of the Dunford-Riesz integral. It is not generally possible to take such a path in case of κ=0.
First,Ut,s is assumed to be bounded for 0≤t,s≤T (equation (10)), and the spectral set of Ut,s is a bounded set in ℂ. Second, for κ satisfying
(18)κ>MeβT,the spectral set of Ut,s+κI is separated with the origin. Consequently, it is possible to take an integral path Γ including the spectral set of Ut,s+κI and excluding the origin. Equation (17) follows from the Dunford-Riesz integral. Furthermore, by adjusting the amplitude of κ, an appropriate integral path always exists independent of t and s. LogUt,s+κI is bounded on X, since Γ is included in the resolvent set of Ut,s+κI [Q.E.D.: Lemma 2].According to this lemma, by introducing nonzeroκ, the logarithm of Ut,s+κI is well-defined without assuming the sectorial property to Ut,s. On the other hand, equation (17) is valid with κ=0 only for limited cases.Theorem 3.
(logarithmic representation of infinitesimal generators). Lett and s satisfy −T≤t,s≤T and Y be a dense subspace of X. For Ut,s defined in Section 2.2, let At∈GX and ∂tUt,s be determined by equations (13) and (14), respectively. If At and Ut,s commute, preinfinitesimal generators At−T≤t≤T are represented by means of the logarithm function; there exists a certain complex number κ≠0 such that
(19)Atus=I+κUs,t∂tLogUt,s+κIus,where us is an element in Y. Note that Ut,s defined in Section 2.2 is assumed to be invertible.
Proof. For Ut,s defined in Section 2.2, operators LogUt,s+κI and LogUt+h,s+κI are well-defined for a certain κ (Lemma 2). The t-differential in a weak sense is formally written by
(20)wlimh→01hLogUt+h,s+κI−LogUt,s+κI=wlimh→01h12πi∫ΓLogλλI−Ut+h,s−κI−1−λI−Ut,s−κI−1dλ=wlimh→012πi∫ΓLogλλI−Ut+h,s−κI−1Ut+h,s−Ut,shλ−Ut,s−κI−1dλ,where Γ, which is possible to be taken independent of t, s, and h for a sufficiently large certain κ, denotes a circle in the resolvent set of both Ut,s+κI and Ut+h,s+κI. A part of the integrand of equation (20) is estimated as
(21)λI−Ut+h,s−κI−1Ut+h,s−Ut,shλI−Ut,s−κI−1vX≤λI−Ut+h,s−κI−1BXUt+h,s−Ut,shλI−Ut,s−κI−1vX,for v∈X. There are two steps to prove the validity of equation (20).
Step 1. The former part of the right hand side of equation (21) satisfies
(22)λI−Ut+h,s−κI−1BX<∞,since λ is taken from the resolvent set of Ut+h,s−κI. In the same way, the operator λ−Ut,s−κ−1 is bounded on X and Y. Then, the continuity of the mapping t⟶λ−Ut,s−κ−1 as for the strong topology follows:
(23)λI−Ut+h,s−κI−1−λI−Ut,s−κI−1BX≤λI−Ut+h,s−κI−1BXUt+h,s−Ut,sλI−Ut,s−κI−1BX.
Step 2. The latter part of the right hand side of equation (21) is estimated as
(24)Ut+h,s−Ut,shλI−Ut,s−κI−1uX=1h∫tt+hAτUτ,sλI−Ut,s−κI−1udτX≤1h∫tt+hAτUτ,sBY,XλI−Ut,s−κI−1BYuYdτ,for ∈Y. Because AτUτ,sBY,X<∞ is true by assumption, the right hand side of equation (24) is finite. Equation (24) shows the uniform boundedness with respect to h; then, the uniform convergence h⟶0 of equation (20) follows. Consequently, the weak limit process h⟶0 for the integrand of equation (8) is justified, as well as the commutation between the limit and the integral.
According to equation (20), interchange of the limit with the integral leads to
(25)∂tLogUt,s+κIu=12πi∫ΓdλLogλλI−Ut,s−κI−1wlimh→0Ut+h,s−Ut,shλI−Ut,s−κI−1u,for ∈Y. Because it is also allowed to interchange At with Ut,s,
(26)∂tLogUt,s+κIu=12πi∫ΓLogλλI−Ut,s−κI−1AtUt,sλI−Ut,s−κI−1dλu=12πi∫ΓLogλλI−Ut,s−κI−2Ut,sdλAtu,for ∈Y. A part of the right hand side is calculated as
(27)12πi∫ΓLogλλI−Ut,s−κI−2Ut,sdλ=12πi∫Γ1λλI−Ut,s−κI−1Ut,sdλ=12πi∫Γ1λλI−Ut,s−κI−1λI−κI−λI−Ut,s−κIdλ=12πi∫Γ1λλI−Ut,s−κI−1dλ−12πi∫ΓκλλI−Ut,s−κI−1dλ−12πi∫Γ1λdλ=12πi∫ΓλI−Ut,s−κI−1dλ−12πi∫ΓκλλI−Ut,s−κI−1dλ=12πi∫ΓλI−Ut,s−κI−1dλ−κUt,s+κI−112πi∫Γ1λUt,s−κI−1λI−Ut,s−κI−1dλ=12πi∫ΓλI−Ut,s−κI−1dλ−κUt,s+κI−112πi∫ΓλI−Ut,s−κI−1dλ−12πi∫Γ1λdλ=I−κUt,s+κI−112πi∫ΓλI−Ut,s−κI−1dλ=I−κUt,s+κI−112πi=r∑n=1∞Ut,snνn+1dν=I−κUt,s+κI−1,due to the integration by parts, where λ−κ=ν=r is a properly chosen circle large enough to include Γ. 2πi−1∫Γλ−1dλ=0 is seen by applying dLogλ/dλ=1/λ. 2πi−1∫ν=r∑n=1∞Ut,snν−n−1dν=I follows from the singularity of ν−n−1.
Consequently,(28)AtusI−κUt,s+κI−1−1∂tLogUt,s+κIus=Ut,s+κIUt,s−1∂tLogUt,s+κIus=I+κUs,t∂tLogUt,s+κIus,is obtained for us∈Y [Q.E.D.: Theorem 3].The meaning of logarithmic representation is examined by focusing on∂tLogUt,s+κI. What is introduced by equation (19) is a kind of resolvent approximation of At(29)∂tLogUt,s+κI=I+κUs,t−1At,in which At is represented by the resolvent operator of Us,t. As seen in the following, it is notable that there is no need to take κ⟶0. This point is different from the usual treatment of resolvent approximations; indeed, it is impossible to take κ⟶0 if the origin is not included in the resolvent set of Ut,s. On the other hand, it is also seen by equation (19) that
(30)∂tLogUt,s+κI=Ut,s+κIκ=0AtUt,s+κI−1,shows a structure of similarity transform, where Ut,s+κκ=0 means Ut,s+κ satisfying a condition κ=0. This asymmetric similarity transform from left and right hand sides are remarkable, and it becomes symmetric if κ=0. A part ∂tLogUt,s+κI plays an essential role in the following discussion.


## 2.1. Banach Algebra

LetA,⋅ be a Banach space (for a textbook, see [16]). A mapping
(1)A×A⟶A,x,y⟶x⋅y,is called a multiplication on A, if it is bilinear and associative. ⋅ is said to be a submultiplicative norm if x⋅y≤xy for each x,y∈A. The Banach space A together with a multiplication and submultiplicative norm is called a Banach algebra.LetX be a Banach space. Denote by BX, the set of all bounded linear operators f:X⟶X. Then, A=BX is an example of a Banach algebra with multiplication as composition and norm defined by
(2)f=supfx:x≤1,where f∈BX. BX is called the operator algebra of X. Let A be a Banach algebra and let X be a Banach space. X is said to be
(i)
A left BanachA-module if there exists a bilinear mapping ⋅:A×X⟶X, a,x⟶a⋅x, called left module action, such that a⋅x≤ax and
(3)ab⋅x=a⋅b⋅x,a,b∈Aandx∈X.(ii)
A right BanachA-module if there exists a bilinear mapping ⋅:A×X⟶X, x,a⟶x⋅a, called right module action, such that x⋅a≤ax and
(4)x⋅ab=x⋅a⋅b,a,b∈Aandx∈X.(iii)
A BanachA-module if it is a left and right Banach A-module and
(5)a⋅x⋅b=a⋅x⋅b,a,b∈Aandx∈X.As an example of the Banach algebra,BX is taken in the following. Then, X is a Banach BX-module under
(6)BX×X⟶X,f,x⟶fx,X×BX⟶X,x,f⟶fx.The Banach space, Banach algebra, and BanachBX-module (BX-module, for short) are the basic concepts in this paper.


## 2.2. Two Parameter Group on Banach Spaces

All the discussion begins with the definition of groups on the Banach spaces that will be generalized to a well-defined semigroup in later sections. LetX,⋅ be a Banach space and BX its Banach algebra. In particular, BX is an example of a Banach algebra.A two parameter group is defined onX. Let X be a Banach space and T∈0,∞. A two parameter group on X is an operator valued mapping t,s⟶Ut,s from −T,T into BX with the semigroup properties:
(7)Ut,rUr,s=Ut,s,r,s,t∈−T,T,(8)Us,s=I,s∈−T,T,and the strong continuity; for eachs∈−T,T and x∈X, the map t⟶Ut,sx is continuous on s,T. Both Ut,s and Us,t are assumed to be well-defined to satisfy
(9)Us,tUt,s=Us,s=I,where Us,t corresponds to the inverse operator of Ut,s. Since Ut,sUs,t=Ut,t=I is also true, the commutation between Ut,s and Us,t follows. Operator Ut,s, which is called the evolution operator in the following, is a generalization of exponential function; indeed, the properties shown in equations (7)–(9) are satisfied by taking Ut,s as et−s. Evolution operator is an abstract concept of exponential function valid for both finite and infinite dimensional Banach spaces. Due to the validity of equation (9), the invertible evolution family is to be associated with some linear evolution equations of hyperbolic type and those of dispersive type. In the same context, the obtained results can be directly applied to some semilinear evolution equations (for a text book, see [17]). For example, the solutions of linear and nonlinear wave equations are written by the evolution operator Ut,s defined above.LetY be a dense Banach subspace of the Banach space X and the topology of Y be stronger than that of X. The space Y⊂X is assumed to be Ut,s-invariant; for any t,s satisfying −T≤t,s≤T,Ut,sY=Y. Following the definition of C0-(semi) group (cf. the assumption H2 in Section 5.3 of Pazy [18] or corresponding discussion in Kato [19, 20]), Ut,s trivially satisfy the boundedness in the present setting; there exist real numbers M and β such that
(10)Ut,sBX≤Meβt,Ut,sBY≤Meβt. that are practically reduced to
(11)Ut,sBX≤MeβT,Ut,sBY≤MeβT,when the interval is restricted to be finite −T,T. Since C0 semigroup theory [21–23], is essentially based on the Laplace transform of operators, the satisfaction of equation (10) is discussed here; Meβt in equation (10) arises from the condition for the existence theorem for the Laplace transforms (for example, see [24]), and M′=MeβT is regarded as a finite real number in the present setting.Next, for the well-definedUt,s, the counterpart of the logarithm in the abstract framework is introduced. There are two concepts associated with the logarithm of operators; one is the infinitesimal generator, and the other is t-differential of Ut,s. These two concepts are connected as follows.Definition 1 (preinfinitesimal generator).
For−T≤t,s≤T, the weak limit
(12)wlimh→0h−1Ut+h,s−Ut,sus=wlimh→0h−1Ut+h,t−IUt,sus,is assumed to exist for certain us, which is an element of a dense subspace Y of X. A linear operator At:Y⟶X is defined by
(13)Atut≔wlimh→0h−1Ut+h,t−Iut,for ut∈Y and −T≤t,s≤T. The operator At for a whole family Ut,s−T≤t,s≤T is called the preinfinitesimal generator.Lett-differential of Ut,s in a weak sense [13] be denoted by
(14)∂tUt,sus=AtUt,sus.Equation (14) is regarded as a differential equation satisfied by ut=Ut,sus that implies a relation between At and the logarithm:
(15)At=∂tUt,sUs,t.The relation betweenAt and the logarithm is discussed in the next Section 2.3. Preinfinitesimal generators are not necessarily infinitesimal generators without assuming a dense property of domain space Y in X. For example, in t-independent cases, an operator At defined by equation (13) is not necessarily a densely defined and closed linear operator, while At must be a densely defined and closed linear operator with its resolvent set included in λ∈ℂ:Reλ>β for At to be the infinitesimal generator. On the other hand, infinitesimal generators are necessarily preinfinitesimal generators. That is, only the exponentiability with a certain ideal domain is valid to the preinfinitesimal generators. The definition of preinfinitesimal generator is useful in terms of providing the algebraic structure. Let a set of preinfinitesimal generators be denoted by GX. It is trivial that BX⊂GX.


## 2.3. Logarithmic Representation of Preinfinitesimal Generator

The logarithmic representation of infinitesimal generator is introduced in order to clarify the structure of infinitesimal generators [8]. The logarithm of Ut,s is defined by the Dunford-Riesz integral [25]. The boundedness of Ut,s on X makes the problem rather easy. Indeed, the boundedness allows us to introduce the translation on the complex plane as a tool to realize the parallel displacement of the entire spectral set. On the other hand, two difficulties inherent to the logarithm
(i)
Singularity of logarithm at the origin(ii)
Multivalued property of the logarithmarise. By introducing a constant κ∈ℂ, the singularity can be handled. This simple treatment is definitely practical to well-define the logarithm of nonsectorial operators. By introducing a principal branch (denoted by “Log”) of the logarithm (denoted by “log”), the multivalued property is handled. Indeed, for any complex number z∈C, a branch of logarithm is defined by
(16)Logz=logz+iargZ,where Z is a complex number chosen to satisfy ∣Z∣=∣z∣, −π<argZ≤π, and argZ=argz+2nπ for a certain integer n.Lemma 2.
(logarithmic representation of operators). Lett and s satisfy 0≤t,s≤T. For a given Ut,s defined in Section 2.2, its logarithm is well defined; there exists a certain complex number κ satisfying
(17)LogUt,s+κI=12πi∫ΓLogλλI−Ut,s−κI−1dλ,where an integral path Γ, which excludes the origin, is a circle in the resolvent set of Ut,s+κI. Here, Γ is independent of t and s. LogUt,s+κI is bounded on X.
Proof. The logarithm Log holds the singularity at the origin so that it is necessary to show a possibility of taking a simple closed curve (integral path) excluding the origin in order to define the logarithm by means of the Dunford-Riesz integral. It is not generally possible to take such a path in case of κ=0.
First,Ut,s is assumed to be bounded for 0≤t,s≤T (equation (10)), and the spectral set of Ut,s is a bounded set in ℂ. Second, for κ satisfying
(18)κ>MeβT,the spectral set of Ut,s+κI is separated with the origin. Consequently, it is possible to take an integral path Γ including the spectral set of Ut,s+κI and excluding the origin. Equation (17) follows from the Dunford-Riesz integral. Furthermore, by adjusting the amplitude of κ, an appropriate integral path always exists independent of t and s. LogUt,s+κI is bounded on X, since Γ is included in the resolvent set of Ut,s+κI [Q.E.D.: Lemma 2].According to this lemma, by introducing nonzeroκ, the logarithm of Ut,s+κI is well-defined without assuming the sectorial property to Ut,s. On the other hand, equation (17) is valid with κ=0 only for limited cases.Theorem 3.
(logarithmic representation of infinitesimal generators). Lett and s satisfy −T≤t,s≤T and Y be a dense subspace of X. For Ut,s defined in Section 2.2, let At∈GX and ∂tUt,s be determined by equations (13) and (14), respectively. If At and Ut,s commute, preinfinitesimal generators At−T≤t≤T are represented by means of the logarithm function; there exists a certain complex number κ≠0 such that
(19)Atus=I+κUs,t∂tLogUt,s+κIus,where us is an element in Y. Note that Ut,s defined in Section 2.2 is assumed to be invertible.
Proof. For Ut,s defined in Section 2.2, operators LogUt,s+κI and LogUt+h,s+κI are well-defined for a certain κ (Lemma 2). The t-differential in a weak sense is formally written by
(20)wlimh→01hLogUt+h,s+κI−LogUt,s+κI=wlimh→01h12πi∫ΓLogλλI−Ut+h,s−κI−1−λI−Ut,s−κI−1dλ=wlimh→012πi∫ΓLogλλI−Ut+h,s−κI−1Ut+h,s−Ut,shλ−Ut,s−κI−1dλ,where Γ, which is possible to be taken independent of t, s, and h for a sufficiently large certain κ, denotes a circle in the resolvent set of both Ut,s+κI and Ut+h,s+κI. A part of the integrand of equation (20) is estimated as
(21)λI−Ut+h,s−κI−1Ut+h,s−Ut,shλI−Ut,s−κI−1vX≤λI−Ut+h,s−κI−1BXUt+h,s−Ut,shλI−Ut,s−κI−1vX,for v∈X. There are two steps to prove the validity of equation (20).
Step 1. The former part of the right hand side of equation (21) satisfies
(22)λI−Ut+h,s−κI−1BX<∞,since λ is taken from the resolvent set of Ut+h,s−κI. In the same way, the operator λ−Ut,s−κ−1 is bounded on X and Y. Then, the continuity of the mapping t⟶λ−Ut,s−κ−1 as for the strong topology follows:
(23)λI−Ut+h,s−κI−1−λI−Ut,s−κI−1BX≤λI−Ut+h,s−κI−1BXUt+h,s−Ut,sλI−Ut,s−κI−1BX.
Step 2. The latter part of the right hand side of equation (21) is estimated as
(24)Ut+h,s−Ut,shλI−Ut,s−κI−1uX=1h∫tt+hAτUτ,sλI−Ut,s−κI−1udτX≤1h∫tt+hAτUτ,sBY,XλI−Ut,s−κI−1BYuYdτ,for ∈Y. Because AτUτ,sBY,X<∞ is true by assumption, the right hand side of equation (24) is finite. Equation (24) shows the uniform boundedness with respect to h; then, the uniform convergence h⟶0 of equation (20) follows. Consequently, the weak limit process h⟶0 for the integrand of equation (8) is justified, as well as the commutation between the limit and the integral.
According to equation (20), interchange of the limit with the integral leads to
(25)∂tLogUt,s+κIu=12πi∫ΓdλLogλλI−Ut,s−κI−1wlimh→0Ut+h,s−Ut,shλI−Ut,s−κI−1u,for ∈Y. Because it is also allowed to interchange At with Ut,s,
(26)∂tLogUt,s+κIu=12πi∫ΓLogλλI−Ut,s−κI−1AtUt,sλI−Ut,s−κI−1dλu=12πi∫ΓLogλλI−Ut,s−κI−2Ut,sdλAtu,for ∈Y. A part of the right hand side is calculated as
(27)12πi∫ΓLogλλI−Ut,s−κI−2Ut,sdλ=12πi∫Γ1λλI−Ut,s−κI−1Ut,sdλ=12πi∫Γ1λλI−Ut,s−κI−1λI−κI−λI−Ut,s−κIdλ=12πi∫Γ1λλI−Ut,s−κI−1dλ−12πi∫ΓκλλI−Ut,s−κI−1dλ−12πi∫Γ1λdλ=12πi∫ΓλI−Ut,s−κI−1dλ−12πi∫ΓκλλI−Ut,s−κI−1dλ=12πi∫ΓλI−Ut,s−κI−1dλ−κUt,s+κI−112πi∫Γ1λUt,s−κI−1λI−Ut,s−κI−1dλ=12πi∫ΓλI−Ut,s−κI−1dλ−κUt,s+κI−112πi∫ΓλI−Ut,s−κI−1dλ−12πi∫Γ1λdλ=I−κUt,s+κI−112πi∫ΓλI−Ut,s−κI−1dλ=I−κUt,s+κI−112πi=r∑n=1∞Ut,snνn+1dν=I−κUt,s+κI−1,due to the integration by parts, where λ−κ=ν=r is a properly chosen circle large enough to include Γ. 2πi−1∫Γλ−1dλ=0 is seen by applying dLogλ/dλ=1/λ. 2πi−1∫ν=r∑n=1∞Ut,snν−n−1dν=I follows from the singularity of ν−n−1.
Consequently,(28)AtusI−κUt,s+κI−1−1∂tLogUt,s+κIus=Ut,s+κIUt,s−1∂tLogUt,s+κIus=I+κUs,t∂tLogUt,s+κIus,is obtained for us∈Y [Q.E.D.: Theorem 3].The meaning of logarithmic representation is examined by focusing on∂tLogUt,s+κI. What is introduced by equation (19) is a kind of resolvent approximation of At(29)∂tLogUt,s+κI=I+κUs,t−1At,in which At is represented by the resolvent operator of Us,t. As seen in the following, it is notable that there is no need to take κ⟶0. This point is different from the usual treatment of resolvent approximations; indeed, it is impossible to take κ⟶0 if the origin is not included in the resolvent set of Ut,s. On the other hand, it is also seen by equation (19) that
(30)∂tLogUt,s+κI=Ut,s+κIκ=0AtUt,s+κI−1,shows a structure of similarity transform, where Ut,s+κκ=0 means Ut,s+κ satisfying a condition κ=0. This asymmetric similarity transform from left and right hand sides are remarkable, and it becomes symmetric if κ=0. A part ∂tLogUt,s+κI plays an essential role in the following discussion.


## 3. Regularized Evolution Operator

### 3.1. Alternative Infinitesimal Generator

The alternative infinitesimal generator is introduced in order to extract bounded parts from the preinfinitesimal generatorA [9]. The operator At∈GX is generally unbounded in X. A bounded operator at,s on X is introduced.Definition 4 (alternative infinitesimal generator).
Letκ be a certain complex number. For a certain vs∈X, the alternative infinitesimal generator ∂tat,s to At∈GX is defined using
(31)at,svs≔LogUt,s+κIvs,on X, where ∂t denotes t − differential in a weak sense.In the present setting assuming the existence of∂tUt,s and therefore At, the operator ∂tat,s exists. According to the logarithmic representation,
(32)Atus=I−κUt,s+κI−1−1∂tat,sus,is obtained. Since at,s is a bounded operator defined by the Dunford-Riesz integral, vs in the definition of alternative infinitesimal generator can be taken from X. Since κ is chosen to separate the spectral set of Ut,s+κI from the origin, the inverse operator of Ut,s+κI always exists, as κ is taken from λ∈ℂ;λ>MeβT.Definition 5 (regularized evolution operator).
The alternative infinitesimal generator∂tat,s generates the regularized evolution operator
(33)eat,s=∑n=0∞at,snn!,which is represented by the convergent power series.The operatoreat,s is regularized in the following sense; the inverse evolution operator e−at,s always exists, if eat,s exists. This fact, which arises from the boundedness of at,s, is true, even if negative time evolution Ut,s−1=Us,t is not well-defined, and only positive time evolution Ut,s is given t>s.It is remarkable that(34)Logeat,s=LogUt,s+κI=at,s,is always satisfied, while
(35)∫I+κUs,t−1Atdt=LogUt,s+κI,is not necessarily satisfied because the limited range of imaginary spectral distribution is necessarily true only for the right hand side. In this sense, at,s corresponds to the extracted bounded part of the infinitesimal generator At. The regularized trajectory in finite/infinite dimensional dynamical systems (for textbooks, see [26, 27]) arises from the regularized evolution operator. Note that, as the well-defined ∂tat,s is not necessary for eat,s to be well-defined, only the well-defined at,s is sufficient for eat,s to be well-defined. This fact essentially simplifies the discussion in applying eat,s in which there is no need to consider weak differential.Using the relation between the logarithm and the exponential functions,(36)Ut,s=eat,s−κI,is valid. It shows a correspondence between eat,s and Ut,s at the level of an evolution operator. One difference is whether the semigroup property is satisfied or not, and another difference is whether the convergence power series representation is always true or not. Meanwhile, at the level of infinitesimal generators, there is a substantial difference between at,s and At. That is, at,s is always bounded on X, while At is not necessarily bounded on X. Because of the boundedness of at,s on X, the inverse operator e−at,s always exists if eat,s exists. One of the essential ideas is to generate eat,s, instead of generating Ut,s.Theorem 6.
(modified semigroup property). Letκ be a certain complex number. For the operator eat,s on X, the semigroup property is replaced with
(37)eat,s−eat,rear,s=κκ+1I−κeat,r+ear,s,(38)eas,s−I=κI.
The inverse relation is replaced with(39)eas,teat,s−eas,s=κeat,s+eas,t−κκ+1I.
In particular, the commutation(40)eas,teat,s−eat,seas,t=0is necessarily valid.
Proof. Substitution of Ut,s=eat,s−κItoUt,rUr,s=Ut,s leads to the following relation:
(41)Ut,rUr,s=eat,s−κI,eat,r−κIear,s−κI=eat,s−κI.where by taking κ with a large ∣κ∣, κ is possible to be taken as common to Ut,s with different t and s. Meanwhile, the replacement of Ut,s=eat,s−κI with Us,s=I leads to the following relation:
(42)eas,s=κ+1I.
That is, forκ≠1, κ+1−1eas.s behaves as the unit operator. Modified version of semigroup property (i.e., (38)) has been proved. The inverse relation (39) follows readily from equation (38). According to equation (15),
(43)eat,t−eat,seas,t=κκ+1I−κeat,s+eas,t,is valid. Combination with another relation
(44)eas,s−eas,teat,s=κκ+1I−κeas,t+eat,s,leads to the commutation:
(45)eat,seas,t−eas,teat,s=0,where eat,t=eas,s=1+κI is utilized [Q.E.D.: Theorem 6].Equations (38) and (39) show the commutativity and violation of semigroup property by eat,s. The right hand sides of equations (38) and (39) are equal to zero for κ=0. These situations correspond to the cases when the semigroup property is satisfied by eat,s, and it is readily seen that the insufficiency of semigroup property arises from the introduction of nonzero κ.The decomposition is obtained by the following structure theorem for the regularized evolution operator. Note that the decomposition ofeat,s also provides a certain relation between the time-discretization and the violation of semigroup property.Theorem 7.
(structure of regularized evolution operator). Letκ be a certain complex number. For a given decomposition s<r1,r2,⋯,rn<t of the interval s,t with n≥2, the operator eat,s on X is represented by
(46)eat,s=eat,rnearn,rn−1⋯ear2,r1ear1,s+κκ+1I−κeat,r1+ear1,s+∑κ=2nκκ+1−κeat,rκ+earκ,rκ−1earκ−1,rκ−2⋯ear2,r1ear1,s,where r0 and rn+1 in the sum are denoted as s=r0 and t=rn+1, respectively.
Proof. According to equation (38), a decomposition
(47)eat,s=eat,r1ear1,s+κκ+1I−κeat,r1+ear1,s,is true. Another decomposition
(48)eat,r1=eat,r2ear2,r1+κκ+1I−κeat,r2ear2,r1,is also true, and then
(49)eat,s=eat,r2ear2,r1+κκ+1I−κeat,r2+ear2,r1ear1,s+κκ+1I−κeat,r1+ear1,s=eat,r2ear2,r1ear1,s+κκ+1I+ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s,follows by sorting based on κ and κκ+1 dependence. Further decomposition shows
(50)eat,r2=eat,r3ear3,r2+κκ+1I−κeat,r3+ear3,r2,and then
(51)eat,s=eat,r2ear2,r1ear1,s+κκ+1I+ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s=eat,r3ear3,r2+κκ+1I−κeat,r3+ear3,r2ear2,r1ear1,s+κκ+1I+ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s=eat,r3ear3,r2ear2,r1ear1,s+κκ+1ear2,r1ear1,s−κeat,r3+ear3,r2ear2,r1ear1,s+κκ+1I+ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s=eat,r3ear3,r2ear2,r1ear1,s+κκ+1I+I+ear2,r1ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s+eat,r3+ear3,r2ear2,r1ear1,s,follows. For a certain n≥2, a constitutional representation is suggested by the deduction:
(52)eat,s=eat,rnearn,rn−1⋯ear2,r1ear1,s+κκ+1I+ear1,s+ear2,r1ear1,s+⋯+earn,rn−1⋯ear2,r1ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s+eat,r3+ear3,r2ear2,r1ear1,s⋯+eat,rnearn,rn−1earn−1,rn−2⋯ear2,r1ear1,s.
Consequently,(53)eat,s=eat,rnearn,rn−1⋯ear2,r1ear1,s+κκ+1I+∑κ=2nearκ−1,rκ−2⋯ear2,r1ear1,s−κeat,r1+ear1,s+∑κ=2neat,rκ+earκ,rκ−1earκ−1,rκ−2⋯ear2,r1ear1,s,is obtained. The statement is proved by sorting terms [Q.E.D.: Theorem 7].Using the regularized evolution operator, the logarithmic representation is readily generalized to the infinitesimal generators of invertible and noninvertible evolution operators. Indeed, according to the proof of Theorem3, only the boundedness of Ut,s on X and the resulting time-interval symmetry is essential.Corollary 8.
(generalized logarithmic representation of infinitesimal generators). Lett and s satisfy −T≤t,s≤T, and Y be a dense subspace of X. For noninvertible Ut,s, Ut,s is defined in Section 2.2 without assuming
(54)Ut,s−1=Us,t,let At∈GX and ∂tUt,s be determined by equations (13) and (14), respectively. If At and Ut,s commute, preinfinitesimal generators At−T≤t≤T are represented by means of the logarithm function; there exists a certain complex number κ≠0 such that
(55)Atus=I−κe−at,s−1∂tLogUt,s+κIus,where us is an element in Y.
Proof. The first line of equation (28) shows the validity of the statement. Indeed, for the logarithmic representation, the invertible property does not play any roles after introducing nonzero κ∈ℂ. In particular, I−κe−at,s−1 is always well-defined for a certain κ [Q.E.D.: Corollary 8].Using the regularized evolution operator,A similarity transform representation (30) for ∂tat,s is written by
(56)∂tat,s=eat,sκ=0Ate−at,s,where the boundedness of at,s allows us to define e−at,s. Due to the boundedness of at,s on X, eat,s is always well-defined by a convergent power series. It leads to the holomorphic property of eat,s. Here is the reason why eat,s is called the regularized evolution operator.In the following, the generalized logarithmic representation of infinitesimal generators is utilized. It enables us to have the logarithmic representation not only for theC0-groups but for the C0-semigroups.


### 3.2. Renormalized Abstract Evolution Equations

Evolution equations are renormalized by means of the alternative infinitesimal generators and regularized evolution operator.Corollary 9.
(renormalized abstract evolution equations). Ifat,s with different t and s are further assumed to commute,
(57)∂teat,svs=∂tat,seat,svsleadingto∂tvt=∂tat,svt,is satisfied for vs∈Y, where ∂t denotes t − differential in a weak sense. This is a linear evolution equation satisfied by vt=eat,svs.
Proof. For the evolution operator
(58)eat,s=∑n=0∞at,snn!,the existence of ∂teat,s is ensured by the existence of ∂tat,s. Using the commutation between at,s with different t and s,
(59)∂tat,sn=nat,sn−1∂tat,s,is true, and the homogeneous-type abstract evolution equation is rephrased as an equation with bounded infinitesimal generator
(60)∂teat,svs=eat,s∂tat,svs,for vs∈Y [Q.E.D.: Corollary 9].This is an abstract evolution equation obtained by the replacementAt with ∂tat,s. Here, one essential idea is to generate eat,s instead of Ut,s; although eat,s is easily defined due to the boundedness of at,s, the general unboundedness of infinitesimal generator ∂tat,s in X is ensured by the similarity transform (56). Under the commutation assumption between At and Ut,s, equation (60) is rephrased as
(61)∂teat,svs=∂tUt,s+κIvs=∂tUt,svs,eat,s∂tat,svs=Ut,s+κI∂tLogUt,s+κIvs=Ut,s+κIUt,s+κI−1Ut,sAtvs=Ut,sAtvs=AtUt,svs,where Theorem 3 is applied. Consequently,
(62)∂tUt,svs=AtUt,svs,is obtained. Note again that eat,s does not satisfy the semigroup property, while Ut,s satisfies it.


### 3.3. Linearized Infinitesimal Generator

The linearity of the semigroup is not assumed in the preceding discussion so that the operatorUt,s can be taken as either linear or nonlinear semigroup. Let us assume a more general situation, in which
(i)
The existence ofUt,s satisfying equation (7) is locally ture for t,s∈−T,T(ii)
The existence of the infinitesimal generator ofUt,s is not clearThis situation corresponds to the situation when only the unique local existence of their solutions the nonlinear partial differential equations is ensured. One of the important application of the renormalized abstract evolution equation is the linearization, which enables to analyze the local profile of nonlinear semigroup.Corollary 10.
(linearized evolution equation). For−T≤t,s≤T, the two parameter group is defined on X. Let either linear or nonlinear semigroup Ut,s defined on a Banach space X satisfy equation (7). Let Ut,sus be the solution of nonlinear equation:
(63)∂tut,s=Fut,s.If the logarithmic representationLogUt,s+κI is true,
(64)∂tvt=I−κUt,s+κI−1−1∂tat,svt,is the linearized equation, where note that at,s includes a parameter κ∈ℂ. If the logarithmic representation is true for κ=0, the infinitesimal generator of linearized problem is simply represented by ∂tat,sκ=0∂tLogUt,s.The condition for obtaining the linearized evolution equation is the locality for the evolution directiont, which leads to the boundedness of the spectral set of Ut,s. The theoretical procedure of obtaining the linearized problem is summarized as follows. For nonzero κ∈ℂ, first, Ut,s is regarded as an exponential function; second, calculating the logarithm of Ut,s; and finally the linearized operator
(65)I−κUt,s+κI−1−1∂tat,s=∂tI−κUt,s+κI−1−1at,s−∂tI−κUt,s+κI−1−1at,s,generates the regularized evolution operator
(66)eat,s−κI.It is more clearly understood by the case ofκ=0,
(67)“Ut,s=eat,sκ=0”leadsto“Logeat,sκ=0=at,sκ=0.”Consequently, the operator-logarithm is regarded as a mapping from “continuous group” to “bounded algebra.” These alternative equations can be used to analyze quasi-linear evolution equations and full-nonlinear evolution equations [28].


#### 3.3.1. Autonomous Case

The regularity results have not been much studied in the Cauchy problem of hyperbolic partial differential equations (for a textbook, see [29]). The regularized evolution operator, which is also applicable to some hyperbolic type equations, is utilized to solve autonomous Cauchy problems.
(68)∂tut=Atutus=us,in X, where At∈GX:Y⟶X is assumed to be an infinitesimal generator of Ut,s satisfying the semigroup property, −T≤t,s≤T is satisfied, Y is a dense subspace of X permitting the representation shown in equation (19), and us is an element of X.As seen in equation (57), under the assumption of commutation, a related Cauchy problem is obtained as
(69)∂tvt,s=∂tat,svt,svs,s=eas,sus,in X, where ∂tat,s=∂tLogUt,s+κI is well-defined. It is possible to solve the rewritten Cauchy problem, and the solution is represented by
(70)vt,s=eat,sus=∑n=0∞at,snn!us,for us∈X (cf. equation (58)).Theorem 11.
Operatoreat,s is holomorphic.
Proof. According to the boundedness of at,s on X (Lemma 2), ∂tneat,s [30] is possible to be represented as
(71)∂tneat,s=12πi∫ΓλneλλI−at,s−1dλ,for a certain κ∈ℂ, where λneλ does not hold any singularity for any finite λ. Following the standard theory of evolution equation,
(72)∂tneat,s≤Cθ,nπtsinθn,is true for a certain constant Cθ,nn=0,1,2,⋯, where θ∈0π/2 and ∣argt∣<π/2 are satisfied (for the detail, e.g., see [22]). It follows that
(73)limt→+0suptn∂tneat,s≤limt→+0suptnCθ,nπtsinθn<∞.
Consequently, for∣z−t∣<tsinθ, the power series expansion
(74)∑n=0∞z−tnn!∂tneat,s,is uniformly convergent in a wider sense. Therefore, eat,s is holomorphic [Q.E.D.: Theorem 11].Theorem 12.
Forus∈X, there exists a unique solution u·∈C−T,T;X for (68) with a convergent power series representation:
(75)ut=Ut,sus=eat,s−κIus=∑n=0∞at,snn!−κIus,where κ is a certain complex number.
Proof. The unique existence follows from the assumption for At. The regularized evolution operator eat,s is holomorphic function (Theorem 11) with the convergent power series representation (equation (58)). By applying Ut,s+κI=eat,s, the solution of the original Cauchy problem is obtained as
(76)ut=eat,s−κIus=∑n=0∞at,sn!n−κIus,for the initial value us∈X. Note that At is not assumed to be a generator of analytic evolution family but only a generator of evolution family [Q.E.D.: Theorem 12].ForIλ denoting the resolvent operator of At, the evolution operator defined by the Yosida approximation is written by
(77)ut=limλ→0exp∫stIλAτdτus,so that more informative representation is provided by Theorem 12 compared to the standard theory based on the Hille-Yosida theorem.


#### 3.3.2. Nonautonomous Case

Series representation in autonomous part leads to the enhancement of the solvability. LetY be a dense subspace of X permitting the representation shown in equation (19), and us is an element of X. The regularized evolution operator is utilized to solved nonautonomous Cauchy problems.
(78)∂tut=Atut+ftus=us,in X, where At∈GX:Y⟶X is assumed to be an infinitesimal generator of Ut,s satisfying the semigroup property and f∈L1−T,T;X is locally Ho¨lder continuous on [−T,T]
(79)ft−fs≤CHt−sγ,for a certain positive constant CH,γ≤1 and −T≤t,s≤T. The solution of nonautonomous problem does not necessarily exist in such a setting (in general, f∈C−T,T;X is necessary).Theorem 13.
Letf∈L1−T,T;X be locally Ho¨lder continuous on −T,T. For us∈X, there exists a unique solution u·∈C−T,T;X for (78) such that
(80)ut=∑n=0∞at,snn!−κIus+∫st∑n=0∞at,τnn!−κIfτdτ,using a certain complex number κ.
Proof. Let us begin with cases with f∈C−T,T;X. The unique existence follows from the standard theory of evolution equation. The representation follows from that of Ut,s and the Duhamel’s principle
(81)ut=Ut,sus+∫stt,τfτdτ=eat,s−κIus+∫steat,τ−κIfτdτ,where the convergent power series representation of eat,s is valid (cf. equation (58)).
(82)ut=eat,s−κIus+∫steat,τ−κIfτdτ,=eat,sus+∫steat,τfτdτ−κIus+∫st−κIfτdτ.
Next, let us consider cases with the locally Ho¨lder continuous ft. According to the linearity of equation (78), it is sufficient to consider the inhomogeneous term. For ϵ satisfying 0<ϵ≪T,
(83)∫st+∈eat,τ−κIfτdτ⟶∫steat,τ−κIfτdτ,is true by taking ϵ⟶0. On the other hand,
(84)At∫st+∈eat,τ−κIfτdτ=∫st+∈AtUt,τfτdτ=∫st+∈AtUt,τfτ−ftdτ+∫st+∈AtUt,τftdτ=∫st+∈AtUt,τfτ−ftdτ−∫st+∈∂τUt,τftdτ=∫st+∈AtUt,τfτ−ftdτ−Ut,t+∈ft+Ut,sft=∫st+∈I+κUs,t∂tat,seat,τ−κIfτ−ftdτ−Ut,t+∈ft+Ut,sft,where ∂τUt,τ=−AτUt,τ is utilized. The last identity is obtained by applying At=I+κUs,t∂tat,s. The Ho¨lder continuity and equation (73) lead to the strong convergence of the right hand of equation (84):
(85)At∫st+∈eat,τ−κIfτdτ⟶∫stI+κUs,t∂tat,seat,τ−κIfτ−ftdτ+Ut,s−Ift,(due to ϵ⟶0) for f∈L10,T;X. At is assumed to be an infinitesimal generator so that At is a closed operator from Y to X. It follows that
(86)∫steat,τ−κIfτdτ∈Y,At∫steat,τ−κIfτdτ=∫stI+κUs,t∂tat,seat,τ−κIfτ−ftdτ+Ut,s−Ift∈X.The right hand side of this equation is strongly continuous on −T,T. Consequently,
(87)∂t∫st+∈eat,τ−κIfτdτ=eat,t+∈−κIft+∈+∫st+∈∂tat,τeat,τfτdτ⟶ft+∫stI−κUτ,t−1AtUt,τ+κIfτdτ=ft+∫stAtUt,τfτdτ=ft+At∫steat,τ−κIfτdτ.
It is seen that∫steat,τ−κIfτdτ satisfies equation (78) and that it is sufficient to assume f∈L10,T;X as Ho¨lder continuous [Q.E.D.: Theorem 13].More simply, the unique solvability of nonautonomous case can be regarded in the context of decomposing the mild solution (for this terminology, see [18]).Corollary 14.
Letf∈L1−T,T;X be locally Ho¨lder continuous on −T,T. For us∈X, there exists a unique solution u⋅∈C−T,T;X for (78) such that
(88)ut=∑n=0∞at,snn!us+∫st∑n=0∞at,snn!fτdτ+κIus−∫stfτdτ,using a certain complex number κ.
Proof. The representation is regarded as
(89)ut=Ut,sus+∫stUt,τfτdτ=eat,s−κIus+∫steat,τ−κIfτdτ=eat,sus+∫steat,τfτdτ−κIus−∫stfτdτ.
The former part in the parenthesis is the mild solution of∂tut=at,sut+ft, and the latter part in another parenthesis is the mild solution of ∂tut=ft. The unique existence of mild solution for the former part is valid for Ho¨lder continuous f∈L10,T;X and that for the latter part is valid for any f∈L1−T,T;X [Q.E.D.: Corollary 14].Corollary 14 shows the meaning of introducing the alternative infinitesimal generator. This result should be compared to the standard theory of evolution equations in which the inhomogeneous termf is assumed to be continuous on −T,T. Consequently, in a purely abstract framework, the maximal regularity effect [31, 32] is found in the solutions of renormalized evolution equations. In this sense, the alternative infinitesimal generator brings about the analytic semigroup theory for nonparabolic evolution equations.


## 3.1. Alternative Infinitesimal Generator

The alternative infinitesimal generator is introduced in order to extract bounded parts from the preinfinitesimal generatorA [9]. The operator At∈GX is generally unbounded in X. A bounded operator at,s on X is introduced.Definition 4 (alternative infinitesimal generator).
Letκ be a certain complex number. For a certain vs∈X, the alternative infinitesimal generator ∂tat,s to At∈GX is defined using
(31)at,svs≔LogUt,s+κIvs,on X, where ∂t denotes t − differential in a weak sense.In the present setting assuming the existence of∂tUt,s and therefore At, the operator ∂tat,s exists. According to the logarithmic representation,
(32)Atus=I−κUt,s+κI−1−1∂tat,sus,is obtained. Since at,s is a bounded operator defined by the Dunford-Riesz integral, vs in the definition of alternative infinitesimal generator can be taken from X. Since κ is chosen to separate the spectral set of Ut,s+κI from the origin, the inverse operator of Ut,s+κI always exists, as κ is taken from λ∈ℂ;λ>MeβT.Definition 5 (regularized evolution operator).
The alternative infinitesimal generator∂tat,s generates the regularized evolution operator
(33)eat,s=∑n=0∞at,snn!,which is represented by the convergent power series.The operatoreat,s is regularized in the following sense; the inverse evolution operator e−at,s always exists, if eat,s exists. This fact, which arises from the boundedness of at,s, is true, even if negative time evolution Ut,s−1=Us,t is not well-defined, and only positive time evolution Ut,s is given t>s.It is remarkable that(34)Logeat,s=LogUt,s+κI=at,s,is always satisfied, while
(35)∫I+κUs,t−1Atdt=LogUt,s+κI,is not necessarily satisfied because the limited range of imaginary spectral distribution is necessarily true only for the right hand side. In this sense, at,s corresponds to the extracted bounded part of the infinitesimal generator At. The regularized trajectory in finite/infinite dimensional dynamical systems (for textbooks, see [26, 27]) arises from the regularized evolution operator. Note that, as the well-defined ∂tat,s is not necessary for eat,s to be well-defined, only the well-defined at,s is sufficient for eat,s to be well-defined. This fact essentially simplifies the discussion in applying eat,s in which there is no need to consider weak differential.Using the relation between the logarithm and the exponential functions,(36)Ut,s=eat,s−κI,is valid. It shows a correspondence between eat,s and Ut,s at the level of an evolution operator. One difference is whether the semigroup property is satisfied or not, and another difference is whether the convergence power series representation is always true or not. Meanwhile, at the level of infinitesimal generators, there is a substantial difference between at,s and At. That is, at,s is always bounded on X, while At is not necessarily bounded on X. Because of the boundedness of at,s on X, the inverse operator e−at,s always exists if eat,s exists. One of the essential ideas is to generate eat,s, instead of generating Ut,s.Theorem 6.
(modified semigroup property). Letκ be a certain complex number. For the operator eat,s on X, the semigroup property is replaced with
(37)eat,s−eat,rear,s=κκ+1I−κeat,r+ear,s,(38)eas,s−I=κI.
The inverse relation is replaced with(39)eas,teat,s−eas,s=κeat,s+eas,t−κκ+1I.
In particular, the commutation(40)eas,teat,s−eat,seas,t=0is necessarily valid.
Proof. Substitution of Ut,s=eat,s−κItoUt,rUr,s=Ut,s leads to the following relation:
(41)Ut,rUr,s=eat,s−κI,eat,r−κIear,s−κI=eat,s−κI.where by taking κ with a large ∣κ∣, κ is possible to be taken as common to Ut,s with different t and s. Meanwhile, the replacement of Ut,s=eat,s−κI with Us,s=I leads to the following relation:
(42)eas,s=κ+1I.
That is, forκ≠1, κ+1−1eas.s behaves as the unit operator. Modified version of semigroup property (i.e., (38)) has been proved. The inverse relation (39) follows readily from equation (38). According to equation (15),
(43)eat,t−eat,seas,t=κκ+1I−κeat,s+eas,t,is valid. Combination with another relation
(44)eas,s−eas,teat,s=κκ+1I−κeas,t+eat,s,leads to the commutation:
(45)eat,seas,t−eas,teat,s=0,where eat,t=eas,s=1+κI is utilized [Q.E.D.: Theorem 6].Equations (38) and (39) show the commutativity and violation of semigroup property by eat,s. The right hand sides of equations (38) and (39) are equal to zero for κ=0. These situations correspond to the cases when the semigroup property is satisfied by eat,s, and it is readily seen that the insufficiency of semigroup property arises from the introduction of nonzero κ.The decomposition is obtained by the following structure theorem for the regularized evolution operator. Note that the decomposition ofeat,s also provides a certain relation between the time-discretization and the violation of semigroup property.Theorem 7.
(structure of regularized evolution operator). Letκ be a certain complex number. For a given decomposition s<r1,r2,⋯,rn<t of the interval s,t with n≥2, the operator eat,s on X is represented by
(46)eat,s=eat,rnearn,rn−1⋯ear2,r1ear1,s+κκ+1I−κeat,r1+ear1,s+∑κ=2nκκ+1−κeat,rκ+earκ,rκ−1earκ−1,rκ−2⋯ear2,r1ear1,s,where r0 and rn+1 in the sum are denoted as s=r0 and t=rn+1, respectively.
Proof. According to equation (38), a decomposition
(47)eat,s=eat,r1ear1,s+κκ+1I−κeat,r1+ear1,s,is true. Another decomposition
(48)eat,r1=eat,r2ear2,r1+κκ+1I−κeat,r2ear2,r1,is also true, and then
(49)eat,s=eat,r2ear2,r1+κκ+1I−κeat,r2+ear2,r1ear1,s+κκ+1I−κeat,r1+ear1,s=eat,r2ear2,r1ear1,s+κκ+1I+ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s,follows by sorting based on κ and κκ+1 dependence. Further decomposition shows
(50)eat,r2=eat,r3ear3,r2+κκ+1I−κeat,r3+ear3,r2,and then
(51)eat,s=eat,r2ear2,r1ear1,s+κκ+1I+ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s=eat,r3ear3,r2+κκ+1I−κeat,r3+ear3,r2ear2,r1ear1,s+κκ+1I+ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s=eat,r3ear3,r2ear2,r1ear1,s+κκ+1ear2,r1ear1,s−κeat,r3+ear3,r2ear2,r1ear1,s+κκ+1I+ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s=eat,r3ear3,r2ear2,r1ear1,s+κκ+1I+I+ear2,r1ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s+eat,r3+ear3,r2ear2,r1ear1,s,follows. For a certain n≥2, a constitutional representation is suggested by the deduction:
(52)eat,s=eat,rnearn,rn−1⋯ear2,r1ear1,s+κκ+1I+ear1,s+ear2,r1ear1,s+⋯+earn,rn−1⋯ear2,r1ear1,s−κeat,r1+ear1,s+eat,r2+ear2,r1ear1,s+eat,r3+ear3,r2ear2,r1ear1,s⋯+eat,rnearn,rn−1earn−1,rn−2⋯ear2,r1ear1,s.
Consequently,(53)eat,s=eat,rnearn,rn−1⋯ear2,r1ear1,s+κκ+1I+∑κ=2nearκ−1,rκ−2⋯ear2,r1ear1,s−κeat,r1+ear1,s+∑κ=2neat,rκ+earκ,rκ−1earκ−1,rκ−2⋯ear2,r1ear1,s,is obtained. The statement is proved by sorting terms [Q.E.D.: Theorem 7].Using the regularized evolution operator, the logarithmic representation is readily generalized to the infinitesimal generators of invertible and noninvertible evolution operators. Indeed, according to the proof of Theorem3, only the boundedness of Ut,s on X and the resulting time-interval symmetry is essential.Corollary 8.
(generalized logarithmic representation of infinitesimal generators). Lett and s satisfy −T≤t,s≤T, and Y be a dense subspace of X. For noninvertible Ut,s, Ut,s is defined in Section 2.2 without assuming
(54)Ut,s−1=Us,t,let At∈GX and ∂tUt,s be determined by equations (13) and (14), respectively. If At and Ut,s commute, preinfinitesimal generators At−T≤t≤T are represented by means of the logarithm function; there exists a certain complex number κ≠0 such that
(55)Atus=I−κe−at,s−1∂tLogUt,s+κIus,where us is an element in Y.
Proof. The first line of equation (28) shows the validity of the statement. Indeed, for the logarithmic representation, the invertible property does not play any roles after introducing nonzero κ∈ℂ. In particular, I−κe−at,s−1 is always well-defined for a certain κ [Q.E.D.: Corollary 8].Using the regularized evolution operator,A similarity transform representation (30) for ∂tat,s is written by
(56)∂tat,s=eat,sκ=0Ate−at,s,where the boundedness of at,s allows us to define e−at,s. Due to the boundedness of at,s on X, eat,s is always well-defined by a convergent power series. It leads to the holomorphic property of eat,s. Here is the reason why eat,s is called the regularized evolution operator.In the following, the generalized logarithmic representation of infinitesimal generators is utilized. It enables us to have the logarithmic representation not only for theC0-groups but for the C0-semigroups.


## 3.2. Renormalized Abstract Evolution Equations

Evolution equations are renormalized by means of the alternative infinitesimal generators and regularized evolution operator.Corollary 9.
(renormalized abstract evolution equations). Ifat,s with different t and s are further assumed to commute,
(57)∂teat,svs=∂tat,seat,svsleadingto∂tvt=∂tat,svt,is satisfied for vs∈Y, where ∂t denotes t − differential in a weak sense. This is a linear evolution equation satisfied by vt=eat,svs.
Proof. For the evolution operator
(58)eat,s=∑n=0∞at,snn!,the existence of ∂teat,s is ensured by the existence of ∂tat,s. Using the commutation between at,s with different t and s,
(59)∂tat,sn=nat,sn−1∂tat,s,is true, and the homogeneous-type abstract evolution equation is rephrased as an equation with bounded infinitesimal generator
(60)∂teat,svs=eat,s∂tat,svs,for vs∈Y [Q.E.D.: Corollary 9].This is an abstract evolution equation obtained by the replacementAt with ∂tat,s. Here, one essential idea is to generate eat,s instead of Ut,s; although eat,s is easily defined due to the boundedness of at,s, the general unboundedness of infinitesimal generator ∂tat,s in X is ensured by the similarity transform (56). Under the commutation assumption between At and Ut,s, equation (60) is rephrased as
(61)∂teat,svs=∂tUt,s+κIvs=∂tUt,svs,eat,s∂tat,svs=Ut,s+κI∂tLogUt,s+κIvs=Ut,s+κIUt,s+κI−1Ut,sAtvs=Ut,sAtvs=AtUt,svs,where Theorem 3 is applied. Consequently,
(62)∂tUt,svs=AtUt,svs,is obtained. Note again that eat,s does not satisfy the semigroup property, while Ut,s satisfies it.


## 3.3. Linearized Infinitesimal Generator

The linearity of the semigroup is not assumed in the preceding discussion so that the operatorUt,s can be taken as either linear or nonlinear semigroup. Let us assume a more general situation, in which
(i)
The existence ofUt,s satisfying equation (7) is locally ture for t,s∈−T,T(ii)
The existence of the infinitesimal generator ofUt,s is not clearThis situation corresponds to the situation when only the unique local existence of their solutions the nonlinear partial differential equations is ensured. One of the important application of the renormalized abstract evolution equation is the linearization, which enables to analyze the local profile of nonlinear semigroup.Corollary 10.
(linearized evolution equation). For−T≤t,s≤T, the two parameter group is defined on X. Let either linear or nonlinear semigroup Ut,s defined on a Banach space X satisfy equation (7). Let Ut,sus be the solution of nonlinear equation:
(63)∂tut,s=Fut,s.If the logarithmic representationLogUt,s+κI is true,
(64)∂tvt=I−κUt,s+κI−1−1∂tat,svt,is the linearized equation, where note that at,s includes a parameter κ∈ℂ. If the logarithmic representation is true for κ=0, the infinitesimal generator of linearized problem is simply represented by ∂tat,sκ=0∂tLogUt,s.The condition for obtaining the linearized evolution equation is the locality for the evolution directiont, which leads to the boundedness of the spectral set of Ut,s. The theoretical procedure of obtaining the linearized problem is summarized as follows. For nonzero κ∈ℂ, first, Ut,s is regarded as an exponential function; second, calculating the logarithm of Ut,s; and finally the linearized operator
(65)I−κUt,s+κI−1−1∂tat,s=∂tI−κUt,s+κI−1−1at,s−∂tI−κUt,s+κI−1−1at,s,generates the regularized evolution operator
(66)eat,s−κI.It is more clearly understood by the case ofκ=0,
(67)“Ut,s=eat,sκ=0”leadsto“Logeat,sκ=0=at,sκ=0.”Consequently, the operator-logarithm is regarded as a mapping from “continuous group” to “bounded algebra.” These alternative equations can be used to analyze quasi-linear evolution equations and full-nonlinear evolution equations [28].


### 3.3.1. Autonomous Case

The regularity results have not been much studied in the Cauchy problem of hyperbolic partial differential equations (for a textbook, see [29]). The regularized evolution operator, which is also applicable to some hyperbolic type equations, is utilized to solve autonomous Cauchy problems.
(68)∂tut=Atutus=us,in X, where At∈GX:Y⟶X is assumed to be an infinitesimal generator of Ut,s satisfying the semigroup property, −T≤t,s≤T is satisfied, Y is a dense subspace of X permitting the representation shown in equation (19), and us is an element of X.As seen in equation (57), under the assumption of commutation, a related Cauchy problem is obtained as
(69)∂tvt,s=∂tat,svt,svs,s=eas,sus,in X, where ∂tat,s=∂tLogUt,s+κI is well-defined. It is possible to solve the rewritten Cauchy problem, and the solution is represented by
(70)vt,s=eat,sus=∑n=0∞at,snn!us,for us∈X (cf. equation (58)).Theorem 11.
Operatoreat,s is holomorphic.
Proof. According to the boundedness of at,s on X (Lemma 2), ∂tneat,s [30] is possible to be represented as
(71)∂tneat,s=12πi∫ΓλneλλI−at,s−1dλ,for a certain κ∈ℂ, where λneλ does not hold any singularity for any finite λ. Following the standard theory of evolution equation,
(72)∂tneat,s≤Cθ,nπtsinθn,is true for a certain constant Cθ,nn=0,1,2,⋯, where θ∈0π/2 and ∣argt∣<π/2 are satisfied (for the detail, e.g., see [22]). It follows that
(73)limt→+0suptn∂tneat,s≤limt→+0suptnCθ,nπtsinθn<∞.
Consequently, for∣z−t∣<tsinθ, the power series expansion
(74)∑n=0∞z−tnn!∂tneat,s,is uniformly convergent in a wider sense. Therefore, eat,s is holomorphic [Q.E.D.: Theorem 11].Theorem 12.
Forus∈X, there exists a unique solution u·∈C−T,T;X for (68) with a convergent power series representation:
(75)ut=Ut,sus=eat,s−κIus=∑n=0∞at,snn!−κIus,where κ is a certain complex number.
Proof. The unique existence follows from the assumption for At. The regularized evolution operator eat,s is holomorphic function (Theorem 11) with the convergent power series representation (equation (58)). By applying Ut,s+κI=eat,s, the solution of the original Cauchy problem is obtained as
(76)ut=eat,s−κIus=∑n=0∞at,sn!n−κIus,for the initial value us∈X. Note that At is not assumed to be a generator of analytic evolution family but only a generator of evolution family [Q.E.D.: Theorem 12].ForIλ denoting the resolvent operator of At, the evolution operator defined by the Yosida approximation is written by
(77)ut=limλ→0exp∫stIλAτdτus,so that more informative representation is provided by Theorem 12 compared to the standard theory based on the Hille-Yosida theorem.


### 3.3.2. Nonautonomous Case

Series representation in autonomous part leads to the enhancement of the solvability. LetY be a dense subspace of X permitting the representation shown in equation (19), and us is an element of X. The regularized evolution operator is utilized to solved nonautonomous Cauchy problems.
(78)∂tut=Atut+ftus=us,in X, where At∈GX:Y⟶X is assumed to be an infinitesimal generator of Ut,s satisfying the semigroup property and f∈L1−T,T;X is locally Ho¨lder continuous on [−T,T]
(79)ft−fs≤CHt−sγ,for a certain positive constant CH,γ≤1 and −T≤t,s≤T. The solution of nonautonomous problem does not necessarily exist in such a setting (in general, f∈C−T,T;X is necessary).Theorem 13.
Letf∈L1−T,T;X be locally Ho¨lder continuous on −T,T. For us∈X, there exists a unique solution u·∈C−T,T;X for (78) such that
(80)ut=∑n=0∞at,snn!−κIus+∫st∑n=0∞at,τnn!−κIfτdτ,using a certain complex number κ.
Proof. Let us begin with cases with f∈C−T,T;X. The unique existence follows from the standard theory of evolution equation. The representation follows from that of Ut,s and the Duhamel’s principle
(81)ut=Ut,sus+∫stt,τfτdτ=eat,s−κIus+∫steat,τ−κIfτdτ,where the convergent power series representation of eat,s is valid (cf. equation (58)).
(82)ut=eat,s−κIus+∫steat,τ−κIfτdτ,=eat,sus+∫steat,τfτdτ−κIus+∫st−κIfτdτ.
Next, let us consider cases with the locally Ho¨lder continuous ft. According to the linearity of equation (78), it is sufficient to consider the inhomogeneous term. For ϵ satisfying 0<ϵ≪T,
(83)∫st+∈eat,τ−κIfτdτ⟶∫steat,τ−κIfτdτ,is true by taking ϵ⟶0. On the other hand,
(84)At∫st+∈eat,τ−κIfτdτ=∫st+∈AtUt,τfτdτ=∫st+∈AtUt,τfτ−ftdτ+∫st+∈AtUt,τftdτ=∫st+∈AtUt,τfτ−ftdτ−∫st+∈∂τUt,τftdτ=∫st+∈AtUt,τfτ−ftdτ−Ut,t+∈ft+Ut,sft=∫st+∈I+κUs,t∂tat,seat,τ−κIfτ−ftdτ−Ut,t+∈ft+Ut,sft,where ∂τUt,τ=−AτUt,τ is utilized. The last identity is obtained by applying At=I+κUs,t∂tat,s. The Ho¨lder continuity and equation (73) lead to the strong convergence of the right hand of equation (84):
(85)At∫st+∈eat,τ−κIfτdτ⟶∫stI+κUs,t∂tat,seat,τ−κIfτ−ftdτ+Ut,s−Ift,(due to ϵ⟶0) for f∈L10,T;X. At is assumed to be an infinitesimal generator so that At is a closed operator from Y to X. It follows that
(86)∫steat,τ−κIfτdτ∈Y,At∫steat,τ−κIfτdτ=∫stI+κUs,t∂tat,seat,τ−κIfτ−ftdτ+Ut,s−Ift∈X.The right hand side of this equation is strongly continuous on −T,T. Consequently,
(87)∂t∫st+∈eat,τ−κIfτdτ=eat,t+∈−κIft+∈+∫st+∈∂tat,τeat,τfτdτ⟶ft+∫stI−κUτ,t−1AtUt,τ+κIfτdτ=ft+∫stAtUt,τfτdτ=ft+At∫steat,τ−κIfτdτ.
It is seen that∫steat,τ−κIfτdτ satisfies equation (78) and that it is sufficient to assume f∈L10,T;X as Ho¨lder continuous [Q.E.D.: Theorem 13].More simply, the unique solvability of nonautonomous case can be regarded in the context of decomposing the mild solution (for this terminology, see [18]).Corollary 14.
Letf∈L1−T,T;X be locally Ho¨lder continuous on −T,T. For us∈X, there exists a unique solution u⋅∈C−T,T;X for (78) such that
(88)ut=∑n=0∞at,snn!us+∫st∑n=0∞at,snn!fτdτ+κIus−∫stfτdτ,using a certain complex number κ.
Proof. The representation is regarded as
(89)ut=Ut,sus+∫stUt,τfτdτ=eat,s−κIus+∫steat,τ−κIfτdτ=eat,sus+∫steat,τfτdτ−κIus−∫stfτdτ.
The former part in the parenthesis is the mild solution of∂tut=at,sut+ft, and the latter part in another parenthesis is the mild solution of ∂tut=ft. The unique existence of mild solution for the former part is valid for Ho¨lder continuous f∈L10,T;X and that for the latter part is valid for any f∈L1−T,T;X [Q.E.D.: Corollary 14].Corollary 14 shows the meaning of introducing the alternative infinitesimal generator. This result should be compared to the standard theory of evolution equations in which the inhomogeneous termf is assumed to be continuous on −T,T. Consequently, in a purely abstract framework, the maximal regularity effect [31, 32] is found in the solutions of renormalized evolution equations. In this sense, the alternative infinitesimal generator brings about the analytic semigroup theory for nonparabolic evolution equations.


## 3.3.1. Autonomous Case

The regularity results have not been much studied in the Cauchy problem of hyperbolic partial differential equations (for a textbook, see [29]). The regularized evolution operator, which is also applicable to some hyperbolic type equations, is utilized to solve autonomous Cauchy problems.
(68)∂tut=Atutus=us,in X, where At∈GX:Y⟶X is assumed to be an infinitesimal generator of Ut,s satisfying the semigroup property, −T≤t,s≤T is satisfied, Y is a dense subspace of X permitting the representation shown in equation (19), and us is an element of X.As seen in equation (57), under the assumption of commutation, a related Cauchy problem is obtained as
(69)∂tvt,s=∂tat,svt,svs,s=eas,sus,in X, where ∂tat,s=∂tLogUt,s+κI is well-defined. It is possible to solve the rewritten Cauchy problem, and the solution is represented by
(70)vt,s=eat,sus=∑n=0∞at,snn!us,for us∈X (cf. equation (58)).Theorem 11.
Operatoreat,s is holomorphic.
Proof. According to the boundedness of at,s on X (Lemma 2), ∂tneat,s [30] is possible to be represented as
(71)∂tneat,s=12πi∫ΓλneλλI−at,s−1dλ,for a certain κ∈ℂ, where λneλ does not hold any singularity for any finite λ. Following the standard theory of evolution equation,
(72)∂tneat,s≤Cθ,nπtsinθn,is true for a certain constant Cθ,nn=0,1,2,⋯, where θ∈0π/2 and ∣argt∣<π/2 are satisfied (for the detail, e.g., see [22]). It follows that
(73)limt→+0suptn∂tneat,s≤limt→+0suptnCθ,nπtsinθn<∞.
Consequently, for∣z−t∣<tsinθ, the power series expansion
(74)∑n=0∞z−tnn!∂tneat,s,is uniformly convergent in a wider sense. Therefore, eat,s is holomorphic [Q.E.D.: Theorem 11].Theorem 12.
Forus∈X, there exists a unique solution u·∈C−T,T;X for (68) with a convergent power series representation:
(75)ut=Ut,sus=eat,s−κIus=∑n=0∞at,snn!−κIus,where κ is a certain complex number.
Proof. The unique existence follows from the assumption for At. The regularized evolution operator eat,s is holomorphic function (Theorem 11) with the convergent power series representation (equation (58)). By applying Ut,s+κI=eat,s, the solution of the original Cauchy problem is obtained as
(76)ut=eat,s−κIus=∑n=0∞at,sn!n−κIus,for the initial value us∈X. Note that At is not assumed to be a generator of analytic evolution family but only a generator of evolution family [Q.E.D.: Theorem 12].ForIλ denoting the resolvent operator of At, the evolution operator defined by the Yosida approximation is written by
(77)ut=limλ→0exp∫stIλAτdτus,so that more informative representation is provided by Theorem 12 compared to the standard theory based on the Hille-Yosida theorem.


## 3.3.2. Nonautonomous Case

Series representation in autonomous part leads to the enhancement of the solvability. LetY be a dense subspace of X permitting the representation shown in equation (19), and us is an element of X. The regularized evolution operator is utilized to solved nonautonomous Cauchy problems.
(78)∂tut=Atut+ftus=us,in X, where At∈GX:Y⟶X is assumed to be an infinitesimal generator of Ut,s satisfying the semigroup property and f∈L1−T,T;X is locally Ho¨lder continuous on [−T,T]
(79)ft−fs≤CHt−sγ,for a certain positive constant CH,γ≤1 and −T≤t,s≤T. The solution of nonautonomous problem does not necessarily exist in such a setting (in general, f∈C−T,T;X is necessary).Theorem 13.
Letf∈L1−T,T;X be locally Ho¨lder continuous on −T,T. For us∈X, there exists a unique solution u·∈C−T,T;X for (78) such that
(80)ut=∑n=0∞at,snn!−κIus+∫st∑n=0∞at,τnn!−κIfτdτ,using a certain complex number κ.
Proof. Let us begin with cases with f∈C−T,T;X. The unique existence follows from the standard theory of evolution equation. The representation follows from that of Ut,s and the Duhamel’s principle
(81)ut=Ut,sus+∫stt,τfτdτ=eat,s−κIus+∫steat,τ−κIfτdτ,where the convergent power series representation of eat,s is valid (cf. equation (58)).
(82)ut=eat,s−κIus+∫steat,τ−κIfτdτ,=eat,sus+∫steat,τfτdτ−κIus+∫st−κIfτdτ.
Next, let us consider cases with the locally Ho¨lder continuous ft. According to the linearity of equation (78), it is sufficient to consider the inhomogeneous term. For ϵ satisfying 0<ϵ≪T,
(83)∫st+∈eat,τ−κIfτdτ⟶∫steat,τ−κIfτdτ,is true by taking ϵ⟶0. On the other hand,
(84)At∫st+∈eat,τ−κIfτdτ=∫st+∈AtUt,τfτdτ=∫st+∈AtUt,τfτ−ftdτ+∫st+∈AtUt,τftdτ=∫st+∈AtUt,τfτ−ftdτ−∫st+∈∂τUt,τftdτ=∫st+∈AtUt,τfτ−ftdτ−Ut,t+∈ft+Ut,sft=∫st+∈I+κUs,t∂tat,seat,τ−κIfτ−ftdτ−Ut,t+∈ft+Ut,sft,where ∂τUt,τ=−AτUt,τ is utilized. The last identity is obtained by applying At=I+κUs,t∂tat,s. The Ho¨lder continuity and equation (73) lead to the strong convergence of the right hand of equation (84):
(85)At∫st+∈eat,τ−κIfτdτ⟶∫stI+κUs,t∂tat,seat,τ−κIfτ−ftdτ+Ut,s−Ift,(due to ϵ⟶0) for f∈L10,T;X. At is assumed to be an infinitesimal generator so that At is a closed operator from Y to X. It follows that
(86)∫steat,τ−κIfτdτ∈Y,At∫steat,τ−κIfτdτ=∫stI+κUs,t∂tat,seat,τ−κIfτ−ftdτ+Ut,s−Ift∈X.The right hand side of this equation is strongly continuous on −T,T. Consequently,
(87)∂t∫st+∈eat,τ−κIfτdτ=eat,t+∈−κIft+∈+∫st+∈∂tat,τeat,τfτdτ⟶ft+∫stI−κUτ,t−1AtUt,τ+κIfτdτ=ft+∫stAtUt,τfτdτ=ft+At∫steat,τ−κIfτdτ.
It is seen that∫steat,τ−κIfτdτ satisfies equation (78) and that it is sufficient to assume f∈L10,T;X as Ho¨lder continuous [Q.E.D.: Theorem 13].More simply, the unique solvability of nonautonomous case can be regarded in the context of decomposing the mild solution (for this terminology, see [18]).Corollary 14.
Letf∈L1−T,T;X be locally Ho¨lder continuous on −T,T. For us∈X, there exists a unique solution u⋅∈C−T,T;X for (78) such that
(88)ut=∑n=0∞at,snn!us+∫st∑n=0∞at,snn!fτdτ+κIus−∫stfτdτ,using a certain complex number κ.
Proof. The representation is regarded as
(89)ut=Ut,sus+∫stUt,τfτdτ=eat,s−κIus+∫steat,τ−κIfτdτ=eat,sus+∫steat,τfτdτ−κIus−∫stfτdτ.
The former part in the parenthesis is the mild solution of∂tut=at,sut+ft, and the latter part in another parenthesis is the mild solution of ∂tut=ft. The unique existence of mild solution for the former part is valid for Ho¨lder continuous f∈L10,T;X and that for the latter part is valid for any f∈L1−T,T;X [Q.E.D.: Corollary 14].Corollary 14 shows the meaning of introducing the alternative infinitesimal generator. This result should be compared to the standard theory of evolution equations in which the inhomogeneous termf is assumed to be continuous on −T,T. Consequently, in a purely abstract framework, the maximal regularity effect [31, 32] is found in the solutions of renormalized evolution equations. In this sense, the alternative infinitesimal generator brings about the analytic semigroup theory for nonparabolic evolution equations.


## 4. Relativistic Formulation of Abstract Evolution Equations

### 4.1. Formalism

The relativistic formulation of abstract evolution equations (55) is introduced to establish an abstract version of the Cole-Hopf transform in Banach spaces and to explain the nonlinear relation between the evolution operator and its infinitesimal generator [11]. The relativistic formulation is introduced for changing the evolution direction, which is necessary to justify the generalized Cole-Hopf transform.In this paper, the logarithmic representation of infinitesimal generator is utilized to formulate the relativistic form of abstract evolution equations. Here, the terminology “relativistic” is used in the sense that there is no especially dominant direction. In particular, the role oft direction (time direction) is not the absolute direction being compared to the other directions: x, y, and z directions (spatial directions) in the standard notation. While the relativistic treatment is associated with the equally valid time-reversal and spatial-reversal symmetries, here the relativistic form to the generalized framework (57) is introduced without assuming the invertible property of evolution operators.Let the standard space-time variablest,x,y,z be denoted by x0,x1,x2,x3 receptively. It is further possible to generalize space-time variables to x0,x1,x2,x3,⋯,xn being valid to general n+1-dimensional space-time. In spite of the standard treatment of abstract evolution equations, the direction of evolution does not necessarily mean time-variable t=x0 in the relativistic formulation of the abstract evolution equations. Consequently, the equal treatment of any direction and the introduction of multidimension are naturally realized by the relativistic formulation.Definition 15 (relativistic form).
For an evolution family of operatorsUxi,ξi−L≤xi,ξi≤L in a Banach space Xi, let Kxi:Yi⟶Xi be the preinfinitesimal generator of Uxi,ξi, where Yi is defence subspace of Xi. The relativistic form of abstract evolution equations is defined as
(90)−∂xiUxi,ξiuξi=KxiUxi,ξiuξi,uξi=uξi,in Xi, where Xi is a functional space consisting of functions with variables xj with 0≤j≤n skipping only j=i. Consequently, the unknown function is represented by uxi=Uxi,ξiuξi for a given initial value uξi∈Xi.Let∂xiUxi,ξiuξi=KxiUxi,ξiuξi evolving for i direction be represented by ∂xkVxk,ξkvξk=KxkVxk,ξkvξk in a certain direction k. For k≠i, let us begin with the abstract Cauchy problem
(91)∂xkVxk,ξkvξk=KxkVxk,ξkvξk,vξk=vξk,in Xk. It is remarkable that even if the evolution operator Uxi,ξi and its infinitesimal generator exist, Vxk,ξk and its infinitesimal generator do not necessarily exist. Those existence should be individually examined for each direction. If Uxi,ξi, Vxk,ξk, and those infinitesimal generators exist, uxi in equation (90) and vxk in equation (91) satisfy the same evolution equation, where the detailed conditions such as initial and boundary conditions can be different depending on the settings of Xi and Xk. For the purpose of introducing the relativistic form with a significance, it is necessary to clarify
(i)
The well-defined (pre-)infinitesimal generator ofVxk,ξk(ii)
The existence ofVxk,ξk (or the corresponding regularized evolution operator)to an unknown direction k. The second one automatically follows if the first one is established. Otherwise equation (91) cannot be regarded as the abstract evolution equations. This issue is examined in generalizing the Cole-Hopf transform.The propagation of singularity should be different if the evolution direction is different. For equations (90) and (91), the evolution direction is not limited to x0. This gives a reason why the formulation shown in equation (90) is called the relativistic form of abstract evolution equations. It means that if invertible evolution operator is obtained for one direction, the evolution operator for the other direction is not necessarily be the invertible. Here is a reason why it is useful to introduce a relativistic form based on the generalized logarithmic representation (cf. Corollary 8).One utility of considering the evolution towards spatial direction is to explain and generalize the Cole-Hopf transform. For this purpose, it is necessary to realize the logarithmic representation of the infinitesimal generators defined in the relativistic form of the abstract evolution equations. That is, for a significant introduction of the relativistic form, it should be introduced together with the logarithmic representation. The condition to obtain the logarithmic representation is stated as follows.Theorem 16.
(relativistic form of logarithmic representation). Leti denote any direction satisfying 0≤i≤n. Let xi and ξi satisfy −L≤xi,ξi≤L and Yi be a dense subspace of a Banach space Xi. A two-parameter evolution family of operators Uxi,ξi−L≤xi,ξi≤L satisfying equation (7) is assumed to exist in a Banach space Xi (i.e., the inverse of Uxi,ξi is not assumed). Under the existence of the preinfinitesimal generator Kxi:Yi⟶Xi of Uxi,ξi for the xi direction, let Uxi,ξi and Kxi commute. The logarithmic representation of infinitesimal generator is obtained; there exists a certain complex number κ≠0 such that
(92)Kxiuξi=I−κe−axi,ξi−1∂xiLogUxi,ξi+κIuξi=I−κe−axi,ξi−1∂xiaxi,ξiuξi,where uξ is an element in Yi, κ is taken from the resolvent set of Uxi,ξi, and axi,ξi=LogUxi,ξi+κI. Note that Uxi,ξi is not assumed to be invertible.
Proof. Different from the proof of Theorem 3, here the similar statement is proved without assuming the invertible property of Uxi,ξi. The key point is that I−κe−axi,ξi−1 exists for a certain κ∈ℂ, even if Uxi,ξi−1 does not exist. In particular, the obtained representation is more generally compared to the one obtained in Ref. [8]. For any Uxi,ξi, operators LogUxi,ξi+κI and LogUxi+h,ξi+κI are well defined for a certain κ. The xi-differential in a weak sense is formally written by
(93)wlimh→01hLogUxi+h,ξi+κI−LogUxi,ξi+κI=wlimh→012πi∫ΓLogλλI−Uxi+h,ξi−κI−1Uxi+h,ξi−Uxi,ξihλI−Uxi,ξi−κI−1dλ,where Γ, which is taken independent of xi, ξi, and h for a sufficiently large certain κ, denotes a circle in the resolvent set of both Ut,s+κI and Ut+h,s+κI.
The discussion, which is the same as that shown in Theorem3, leads to
(94)∂tLogUxi,ξi+κIu=12πi∫ΓdλLogλλI−Uxi,ξi−κI−1wlimh→0Uxi+h,ξi−Uxi,ξihλI−Uxi,ξi−κI−1u,for u∈Yi. Because it is allowed to interchange Kxi with Uxi,ξi,
(95)∂tLogUxi,ξi+κIu=12πi∫ΓLogλλI−Uxi,ξi−κI−2Ut,sdλKxiu,for u∈Yi, where wlimh→0Uxi+h,ξi−Uxi,ξi/h means the preinfinitesimal generator Kxi itself. A part of the right hand side is calculated as
(96)12πi∫ΓLogλλI−Uxi,ξi−κI−2Uxi,ξidλ=I−κUt,s+κI−1,due to the integration by parts, where the details of procedure is essentially the same as Ref. [8]. It leads to
(97)Kxiuξi=I−κe−axi,ξi−1∂xiLogUxi,ξi+κIuξi,for uξi∈Yi. It is notable that Uxi,ξi+κI−1 is always well-defined for any κ taken from the resolvent set of Uxi,ξi, even if Uxi,ξi=Uxi,ξi−1 does not exist [Q.E.D.: Theorem 16].Under the existence of logarithmic representation forKxi, the related concepts such as
(1)
Alternative infinitesimal generator:∂xiaxi,ξi=∂xiLogUxi,ξi+κI(2)
Regularized evolution operator:eaxi,ξi=Uxi,ξi+κI(3)
Renormalized abstract evolution equation:∂xiu~=∂xiaxi,ξiu~ are similarly well-defined in the relativistic framework


### 4.2. Generalization of the Cole-Hopf Transform

Now it is ready for establishing the general version of the Cole-Hopf transform. It corresponds to an application example of relativistic formulation provided. The Cole-Hopf transform [33–37] is a concept bridging the linearity and the nonlinearity. In the following, such a linear-nonlinear conversion relation is found within the relation between the infinitesimal generators and the generated semigroups. For t∈ℝ+ and x∈ℝ, the Cole-Hopf transform reads
(98)ψt,x=−2μ−1/2∂xlogut,x,where ut,x denotes the solution of linear equation and ψt,x is the solution of transformed nonlinear equation. On the other hand, e.g., for t∈ℝ and x∈ℝn, the logarithmic representation of infinitesimal generator
(99)Kxiuξi=I+κUxi,ξi∂xiLogUxi,ξi+κIuξi,has been obtained in the abstract framework, where Uxi,ξi denotes the evolution operator and Kxi is its infinitesimal generator. By taking a specific case with κ=0, the similarity between them is clear. That is, the process of obtaining infinitesimal generators from evolution operators is expected to be related to the emergence of nonlinearity.Based on the logarithmic representation of infinitesimal generators obtained in the Banach spaces, the Cole-Hopf transform is generalized in the following sense:(i)
The linear equation is not necessarily the heat equation(ii)
The spatial dimension of the equations is not limited to 1(iii)
The variable in the transform is not limited to a spatial variablexwhere in order to realize these features, the relativistic formulation of abstract evolution equation is newly introduced. Since the logarithmic representation shows a relation between an evolution operator and its infinitesimal generator, the correspondence to the Cole-Hopf transform means a possible appearance of nonlinearity in the process of defining an infinitesimal generator from the evolution operator. The next theorem follows.Theorem17.
(generalization of the Cole-Hopf transform). Leti be an integer satisfying 0≤i≤n and Y be a dense subspace of a Banach space X. Let an invertible evolution family Uxi,ξi0≤xi,ξi≤L be generated by Axi for 0≤xi,ξi≤L in a Banach space X. Uxi,ξi and Axi are assumed to commute. For any uξi∈Y⊂X, the logarithmic representation
(100)AxiUxi,ξiuξi=eaxi,ξi∂xiLogUxi,ξi+κIuξi,is the generalization of the Cole-Hopf transform, where the logarithmic representation is obtained in a general Banach space framework, κ≠0 is a complex number, and where axi,ξi=LogUxi,ξi+κI. In particular, if Uxi,ξiuξi−1 exists for a given interval 0≤xi,ξi≤L, its normalization
(101)Axi=I−κe−axi,ξi∂xiLogUxi,ξi+κI,defined in X corresponds to ψt,x.
Proof. The proof consists of five steps.
Step 1. Formulation. It is necessary to recognize the evolution direction of the heat equation as x, because the derivative on the spatial direction x is considered in the Cole-Hopf transform. The Cole-Hopf transform acts on one-dimensional heat equation
(102)∂x2ut,x−μ1/2∂tut,x=0,t∈0,∞,x∈−L,L,ut,−L=ut,L=0,t∈0,∞,u0,x=u0x,x∈−L,L,where μ is a real positive number and the hypoelliptic property of parabolic evolution equation is true. The first equation of (102) is hypoelliptic; for an open set U⊂−∞,∞×−L,L, u∈C∞U follows from ∂x2−μ1/2∂tu∈C∞U. Equation (102) is well-posed in C∞0,∞×C∞−L,L so that μ1/2∂t is the infinitesimal generator in C∞−L,L. The spaces C1−L,L and C∞−L,L are dense in Lp−L,L with 1≤p<∞. The solution is represented by
(103)ut,x=Utu0,where Ut is a semigroup generated by μ−1/2∂x2 under the Dirichlet-zero boundary condition.
By changing the evolution direction fromt to x, the heat equation
(104)∂x2ut,x−μ1/2∂tut,x=0,t∈0,∞,x∈−L,L,(105)ut,−L=v0t,∂xut,−L=v1t,t∈0,∞,(106)u0,x=0x∈−L,L,is considered for x direction, where v0t and v1t are given initial functions. To establish the existence of semigroup for the x direction, it is sufficient to consider the generation of semigroup in L2−∞,∞ by generalizing t-interval from 0,∞ to −∞,∞. The Fourier transform leads to
(107)∂x2u~−iμ1/2ωu~=0u~ω,0=v~0ω,∂xu~ω,0=v~1ω,where ω is a real number. Indeed, the following transforms
(108)ut,x=12π∫−∞∞u~∫−∞∞u~ω,xeiωtdω,ut,−L=12π∫−∞∞u~ω,−Leiωtdω,∂xut,−L=12π∫−∞∞∂xu~ω,−Leiωtdω,are implemented. By solving the characteristic equation λ2−iμ1/2ω−0, the Fourier transformed solution of (105) is
(109)u~ω,x=v~0ω−i−iμ1/2ω−1/2v~1ω2e+iμ1/2ω1/2x+v~0ω+i−iμ1/2ω−1/2v~1ω2e+iμ1/2ω1/2x,where
(110)u~ω,0=v~0ω−i−iμ1/2ω1/2v~1ω2+v~0ω+i−iμ1/2ω1/2v~1ω2v~0ω,∂xu~ω,0=iμ1/2ω1/2v~0ω−i−iμ1/2ω−1/2v~1ω2−v~0ω+i−iμ1/2ω−1/2v~1ω2=v~1ω.
Meanwhile, based on the relativistic treatment, one-dimensional heat equation(111)∂x2u−μ1/2∂tu=0,is written as
(112)∂xuv−0Ifμ1/2∂t0uv=0.
Let a linear operatorA be defined by
(113)A==0Iμ1/2∂t0,in L2−∞,∞×L2−∞,∞ and the domain space of A be
(114)DA=H1−∞,∞×L2−∞,∞,where H1−∞,∞=ut∈L2−∞,∞;u′t∈L2−∞,∞,u0=0 is a Sobolev space. The Fourier transform means that the diagonalization of A is equal to
(115)A~=μ1/2∂t1/200−μ1/2∂t−1/2.
In this context, the master equation of the problem (105) is reduced to the abstract evolution equation
(116)∂xu~v~−A~u~v~=0,in L2−∞,∞×L2−∞,∞, where
(117)u~=u−i−iμ1/2ω−1/2v2,v~=u+i−iμ1/2ω−1/2v2.
It suggests that the evolution operator of equation (105) is generated by
(118)±μ1/2∂t1/2,so that it is sufficient to show ±μ1/2∂t1/2, as the infinitesimal generator. Note that the operator A~ is not necessarily a generator of analytic semigroup, because the propagation of singularity should be different if the evolution direction is different. Consequently, the existence of semigroup for (116) in the x direction is reduced to show ±μ1/2∂t1/2, as the infinitesimal generator in L2−∞,∞. In the following, the property of μ1/2∂t is discussed in the second step, and the fractional power of μ1/2∂t1/2 is studied in the third step.
Step 2. First order differential operator. The following lemma is proved in this step.
Lemma 18. The operator μ1/2∂t with the domain H1−∞,∞ is the infinitesimal generator in L2−∞,∞.
Proof of Lemma 18 Letλ be a complex number satisfying Reλ>0. First, the existence of λ−μ1/2∂t−1 is examined. Let ft be included in L2−∞,∞. Because
(119)λ−μ1/2∂tu=f,in one-dimensional interval −∞,∞ is a first-order ordinary differential equation with a constant coefficient, and the global-in-t solution necessarily exists for a given u0=u0∈ℂ. That is, λ/μ1/2 is included in the resolvent set of ∂t for an arbitrary complex number λ so that λ−μ1/2∂t−1 is concluded to be well-defined in H1−∞,∞.
Second, the resolvent operatorλ−μ1/2∂t−1 is estimated from the above. Since λ/μ1/2 is included in the resolvent set of ∂t, it is readily seen that λ−μ1/2∂t−1 is a bounded operator on L2−∞,∞. More precisely, let us consider equation (119) being equivalent to
(120)∂tut=λμ1/2ut−ftμ1/2.
If the inhomogeneous term satisfiesft∈L2−∞,∞,
(121)ut=−1μ1/2∫t∞exp−λμ1/2s−tfsds,satisfies equation (120). According to the Schwarz inequality,
(122)∫−∞∞ut2dt=∫−∞∞1μ1/2∫t∞exp−λμ1/2s−tfsds2dt≤1μ∫−∞∞∫t∞exp−Reλ2μ1/2s−texp−Reλ2μ1/2s−tfsds2dt≤1μ∫−∞∞∫t∞exp−Reλμ1/2s−tds∫t∞exp−Reλμ1/2s−tfs2dsdt,is obtained, and the equality
(123)∫t∞exp−Reλμ1/2s−tds=∫0∞exp−Reλμ1/2sds=μ1/2Reλ,is positive if Reλ>0 is satisfied. Its application leads to
(124)∫−∞∞ut2dt≤1μμ1/2Reλ∫−∞∞∫t∞exp−Reλμ1/2s−tfs2dsdt=1μμ1/2Reλ∫−∞∞∫−∞sexp−Reλμ1/2s−tdtfs2ds.
Further application of the equality(125)∫−∞sexp−Reλμ1/2s−tdt=∫−∞0expReλμ1/2tdt=μ1/2Reλ,results in
(126)∫−∞∞ut2dt≤1μμ1/2Reλ∫−∞∞μ1/2Reλfs2ds=1Reλ2∫−∞∞fs2ds,for Reλ>0, and therefore,
(127)λI−μ1/2∂t−1ftL2−∞,∞≤1ReλftL2−∞,∞,follows. That is, for Reλ>0,
(128)λI−μ1/2∂t−1≤1/Reλ,is valid. The surjective property of λI−μ1/2∂t is seen by the unique existence of solutions u∈L2−∞,∞ for the Cauchy problem of equation (120).
A semigroup is generated by taking a subset of the complex plane as(129)Ω=λ∈ℂ;λ=λ¯,where Ω is included in the resolvent set of μ1/2∂t. For λ∈Ω,λI−μ1/2∂t−1 exists, and
(130)λI−μ1/2∂t−n≤1/Reλn,is obtained. Consequently, according to the Lumer-Phillips theorem [38] for the generation of quasi contraction semigroup, μ1/2∂t is confirmed to be an infinitesimal generator in L2−∞,∞, and the unique existence of global-in-x weak solution follows [Q.E.D.: Lemma 18].
The semigroup generated byμ1/2∂t is represented by
(131)exphμ1/2∂tut=ut+μ1/2h,−∞<h<∞,so that the group is actually generated by μ1/2∂t. Indeed, the similar estimate as equation (130) can be obtained for Reλ<0 with λ−μ1/2u=f in which the solution u is represented by
(132)ut=−1μ1/2∫−∞texp−λμ1/2s−tfsds,that should be compared to equation (121).
Step 3. Fractional powers of operator. The following lemma is proved in this step.
Lemma 19.  For 0<α<1, the operator μ1/2∂tα is the infinitesimal generator in L2−∞,∞.
Proof of Lemma 19 According to Lemma 18,μ1/2∂t is the infinitesimal generator in L2−∞,∞. For an infinitesimal generator μ1/2∂t in L2−∞,∞, let the one-parameter semigroup generated by μ1/2∂t be denoted by Vx. An infinitesimal generator μ1/2∂t is a closed linear operator in L2−∞,∞. Its fractional power
(133)μ1/2∂tα,0<α<1,has been confirmed to be well-defined by S. Bochner [39] and R.S. Phillips [40] as the infinitesimal generator of semigroup (cf. K. Yosida [41]):
(134)Wxw0=∫0∞Vxw0dγλ,for w0∈L2−∞,∞, where Wx is the semigroup for x direction. The measure dγλ≥0 is defined through the Laplace integral
(135)exp−tkα=∫0∞exp−λkdγλ,where t,k>0 is satisfied [Q.E.D.: Lemma 19].
By takingα=1/2, μ1/2∂t1/2 is confirmed to be an infinitesimal generator in L2−∞,∞. Because Ω is included in the resolvent set of −μ1/2∂t1/2, it is readily seen that −μ1/2∂t1/2 is also an infinitesimal generator in L2−∞,∞.
Step 4. Abstract form of the Cole-Hopf transform. As in the original derivation of the Cole-Hopf transform, the solution of heat equation wt,x=Wx,ξwξt solved along the x direction permits its logarithm function. The abstract case of the original Cole-Hopf transform is included in the description of the logarithmic representation (100). Indeed, let an invertible evolution family Wx,ξ0≤x,ξ≤L be generated by Ax for 0≤x,ξ≤L. According to Lemma 19, the logarithmic representation of relativistic form (99) is obtained in this case as
(136)Axwξ=I+κWξ,x∂xLogWx,ξ+κIwξ,and hence as
(137)Axwx,ξwξ=κI+Wx,ξ∂xLogWx,ξ+κIwξ,using the commutation assumption. The nonlinear Anzatz −2μ−1/2∂xut,xut,x−1 of the Burgers’ equation
(138)ψt,x=−2μ−1/2∂xlogut,x=−2μ−1/2∂xut,xut,x−1,is essentially represented by
(139)−2μ−1/2AxWx,ξWx,ξ−1=−2μ−1/2κI+Wx,ξ∂xLogWx,ξ+κIWx,ξ−1=−2μ−1/2κWξ,x+I∂xLogWx,ξ+κI,in the abstract form. The similarity between equation (101) and the standard definition of operator norm is clear. In particular, the evolution direction is generalized from x to xi in equation (101).
Step 5. Generalization property. Equation (100) is the generalization of the Cole-Hopf transform. According to the introduction of nonzero κ in the abstract form, the applicability is significantly increased so that the linear equation is not necessarily the heat equation. According to the abstract nature of the logarithmic representation, linear and nonlinear equations are not necessarily considered in the one-spatial dimension. According to the relativistic treatment, the transformed variable is not limited to the spatial variables [Q.E.D.: Theorem 17].The generalized Cole-Hopf transform (100) shows that the nonlinearity of semigroup can appear simply by altering the evolution direction under a suitable identification between the infinitesimal generator and the evolution operator. In this sense, equation (90) is regarded as a local-in-xi linearized equation, if Uxi,ξi is a nonlinear semigroup (semigroup related to the nonlinear equations). Furthermore, the generalized Cole-Hopf transform (100) suggests that the relation between evolution operator and its infinitesimal generator corresponds essentially to the transform between linearity and nonlinearity. In the same context of generalizing Miura transform between the Korteweg-de-Vries and the modified Korteweg-de-Vries equations, the logarithmic representation is utilized [15].


## 4.1. Formalism

The relativistic formulation of abstract evolution equations (55) is introduced to establish an abstract version of the Cole-Hopf transform in Banach spaces and to explain the nonlinear relation between the evolution operator and its infinitesimal generator [11]. The relativistic formulation is introduced for changing the evolution direction, which is necessary to justify the generalized Cole-Hopf transform.In this paper, the logarithmic representation of infinitesimal generator is utilized to formulate the relativistic form of abstract evolution equations. Here, the terminology “relativistic” is used in the sense that there is no especially dominant direction. In particular, the role oft direction (time direction) is not the absolute direction being compared to the other directions: x, y, and z directions (spatial directions) in the standard notation. While the relativistic treatment is associated with the equally valid time-reversal and spatial-reversal symmetries, here the relativistic form to the generalized framework (57) is introduced without assuming the invertible property of evolution operators.Let the standard space-time variablest,x,y,z be denoted by x0,x1,x2,x3 receptively. It is further possible to generalize space-time variables to x0,x1,x2,x3,⋯,xn being valid to general n+1-dimensional space-time. In spite of the standard treatment of abstract evolution equations, the direction of evolution does not necessarily mean time-variable t=x0 in the relativistic formulation of the abstract evolution equations. Consequently, the equal treatment of any direction and the introduction of multidimension are naturally realized by the relativistic formulation.Definition 15 (relativistic form).
For an evolution family of operatorsUxi,ξi−L≤xi,ξi≤L in a Banach space Xi, let Kxi:Yi⟶Xi be the preinfinitesimal generator of Uxi,ξi, where Yi is defence subspace of Xi. The relativistic form of abstract evolution equations is defined as
(90)−∂xiUxi,ξiuξi=KxiUxi,ξiuξi,uξi=uξi,in Xi, where Xi is a functional space consisting of functions with variables xj with 0≤j≤n skipping only j=i. Consequently, the unknown function is represented by uxi=Uxi,ξiuξi for a given initial value uξi∈Xi.Let∂xiUxi,ξiuξi=KxiUxi,ξiuξi evolving for i direction be represented by ∂xkVxk,ξkvξk=KxkVxk,ξkvξk in a certain direction k. For k≠i, let us begin with the abstract Cauchy problem
(91)∂xkVxk,ξkvξk=KxkVxk,ξkvξk,vξk=vξk,in Xk. It is remarkable that even if the evolution operator Uxi,ξi and its infinitesimal generator exist, Vxk,ξk and its infinitesimal generator do not necessarily exist. Those existence should be individually examined for each direction. If Uxi,ξi, Vxk,ξk, and those infinitesimal generators exist, uxi in equation (90) and vxk in equation (91) satisfy the same evolution equation, where the detailed conditions such as initial and boundary conditions can be different depending on the settings of Xi and Xk. For the purpose of introducing the relativistic form with a significance, it is necessary to clarify
(i)
The well-defined (pre-)infinitesimal generator ofVxk,ξk(ii)
The existence ofVxk,ξk (or the corresponding regularized evolution operator)to an unknown direction k. The second one automatically follows if the first one is established. Otherwise equation (91) cannot be regarded as the abstract evolution equations. This issue is examined in generalizing the Cole-Hopf transform.The propagation of singularity should be different if the evolution direction is different. For equations (90) and (91), the evolution direction is not limited to x0. This gives a reason why the formulation shown in equation (90) is called the relativistic form of abstract evolution equations. It means that if invertible evolution operator is obtained for one direction, the evolution operator for the other direction is not necessarily be the invertible. Here is a reason why it is useful to introduce a relativistic form based on the generalized logarithmic representation (cf. Corollary 8).One utility of considering the evolution towards spatial direction is to explain and generalize the Cole-Hopf transform. For this purpose, it is necessary to realize the logarithmic representation of the infinitesimal generators defined in the relativistic form of the abstract evolution equations. That is, for a significant introduction of the relativistic form, it should be introduced together with the logarithmic representation. The condition to obtain the logarithmic representation is stated as follows.Theorem 16.
(relativistic form of logarithmic representation). Leti denote any direction satisfying 0≤i≤n. Let xi and ξi satisfy −L≤xi,ξi≤L and Yi be a dense subspace of a Banach space Xi. A two-parameter evolution family of operators Uxi,ξi−L≤xi,ξi≤L satisfying equation (7) is assumed to exist in a Banach space Xi (i.e., the inverse of Uxi,ξi is not assumed). Under the existence of the preinfinitesimal generator Kxi:Yi⟶Xi of Uxi,ξi for the xi direction, let Uxi,ξi and Kxi commute. The logarithmic representation of infinitesimal generator is obtained; there exists a certain complex number κ≠0 such that
(92)Kxiuξi=I−κe−axi,ξi−1∂xiLogUxi,ξi+κIuξi=I−κe−axi,ξi−1∂xiaxi,ξiuξi,where uξ is an element in Yi, κ is taken from the resolvent set of Uxi,ξi, and axi,ξi=LogUxi,ξi+κI. Note that Uxi,ξi is not assumed to be invertible.
Proof. Different from the proof of Theorem 3, here the similar statement is proved without assuming the invertible property of Uxi,ξi. The key point is that I−κe−axi,ξi−1 exists for a certain κ∈ℂ, even if Uxi,ξi−1 does not exist. In particular, the obtained representation is more generally compared to the one obtained in Ref. [8]. For any Uxi,ξi, operators LogUxi,ξi+κI and LogUxi+h,ξi+κI are well defined for a certain κ. The xi-differential in a weak sense is formally written by
(93)wlimh→01hLogUxi+h,ξi+κI−LogUxi,ξi+κI=wlimh→012πi∫ΓLogλλI−Uxi+h,ξi−κI−1Uxi+h,ξi−Uxi,ξihλI−Uxi,ξi−κI−1dλ,where Γ, which is taken independent of xi, ξi, and h for a sufficiently large certain κ, denotes a circle in the resolvent set of both Ut,s+κI and Ut+h,s+κI.
The discussion, which is the same as that shown in Theorem3, leads to
(94)∂tLogUxi,ξi+κIu=12πi∫ΓdλLogλλI−Uxi,ξi−κI−1wlimh→0Uxi+h,ξi−Uxi,ξihλI−Uxi,ξi−κI−1u,for u∈Yi. Because it is allowed to interchange Kxi with Uxi,ξi,
(95)∂tLogUxi,ξi+κIu=12πi∫ΓLogλλI−Uxi,ξi−κI−2Ut,sdλKxiu,for u∈Yi, where wlimh→0Uxi+h,ξi−Uxi,ξi/h means the preinfinitesimal generator Kxi itself. A part of the right hand side is calculated as
(96)12πi∫ΓLogλλI−Uxi,ξi−κI−2Uxi,ξidλ=I−κUt,s+κI−1,due to the integration by parts, where the details of procedure is essentially the same as Ref. [8]. It leads to
(97)Kxiuξi=I−κe−axi,ξi−1∂xiLogUxi,ξi+κIuξi,for uξi∈Yi. It is notable that Uxi,ξi+κI−1 is always well-defined for any κ taken from the resolvent set of Uxi,ξi, even if Uxi,ξi=Uxi,ξi−1 does not exist [Q.E.D.: Theorem 16].Under the existence of logarithmic representation forKxi, the related concepts such as
(1)
Alternative infinitesimal generator:∂xiaxi,ξi=∂xiLogUxi,ξi+κI(2)
Regularized evolution operator:eaxi,ξi=Uxi,ξi+κI(3)
Renormalized abstract evolution equation:∂xiu~=∂xiaxi,ξiu~ are similarly well-defined in the relativistic framework


## 4.2. Generalization of the Cole-Hopf Transform

Now it is ready for establishing the general version of the Cole-Hopf transform. It corresponds to an application example of relativistic formulation provided. The Cole-Hopf transform [33–37] is a concept bridging the linearity and the nonlinearity. In the following, such a linear-nonlinear conversion relation is found within the relation between the infinitesimal generators and the generated semigroups. For t∈ℝ+ and x∈ℝ, the Cole-Hopf transform reads
(98)ψt,x=−2μ−1/2∂xlogut,x,where ut,x denotes the solution of linear equation and ψt,x is the solution of transformed nonlinear equation. On the other hand, e.g., for t∈ℝ and x∈ℝn, the logarithmic representation of infinitesimal generator
(99)Kxiuξi=I+κUxi,ξi∂xiLogUxi,ξi+κIuξi,has been obtained in the abstract framework, where Uxi,ξi denotes the evolution operator and Kxi is its infinitesimal generator. By taking a specific case with κ=0, the similarity between them is clear. That is, the process of obtaining infinitesimal generators from evolution operators is expected to be related to the emergence of nonlinearity.Based on the logarithmic representation of infinitesimal generators obtained in the Banach spaces, the Cole-Hopf transform is generalized in the following sense:(i)
The linear equation is not necessarily the heat equation(ii)
The spatial dimension of the equations is not limited to 1(iii)
The variable in the transform is not limited to a spatial variablexwhere in order to realize these features, the relativistic formulation of abstract evolution equation is newly introduced. Since the logarithmic representation shows a relation between an evolution operator and its infinitesimal generator, the correspondence to the Cole-Hopf transform means a possible appearance of nonlinearity in the process of defining an infinitesimal generator from the evolution operator. The next theorem follows.Theorem17.
(generalization of the Cole-Hopf transform). Leti be an integer satisfying 0≤i≤n and Y be a dense subspace of a Banach space X. Let an invertible evolution family Uxi,ξi0≤xi,ξi≤L be generated by Axi for 0≤xi,ξi≤L in a Banach space X. Uxi,ξi and Axi are assumed to commute. For any uξi∈Y⊂X, the logarithmic representation
(100)AxiUxi,ξiuξi=eaxi,ξi∂xiLogUxi,ξi+κIuξi,is the generalization of the Cole-Hopf transform, where the logarithmic representation is obtained in a general Banach space framework, κ≠0 is a complex number, and where axi,ξi=LogUxi,ξi+κI. In particular, if Uxi,ξiuξi−1 exists for a given interval 0≤xi,ξi≤L, its normalization
(101)Axi=I−κe−axi,ξi∂xiLogUxi,ξi+κI,defined in X corresponds to ψt,x.
Proof. The proof consists of five steps.
Step 1. Formulation. It is necessary to recognize the evolution direction of the heat equation as x, because the derivative on the spatial direction x is considered in the Cole-Hopf transform. The Cole-Hopf transform acts on one-dimensional heat equation
(102)∂x2ut,x−μ1/2∂tut,x=0,t∈0,∞,x∈−L,L,ut,−L=ut,L=0,t∈0,∞,u0,x=u0x,x∈−L,L,where μ is a real positive number and the hypoelliptic property of parabolic evolution equation is true. The first equation of (102) is hypoelliptic; for an open set U⊂−∞,∞×−L,L, u∈C∞U follows from ∂x2−μ1/2∂tu∈C∞U. Equation (102) is well-posed in C∞0,∞×C∞−L,L so that μ1/2∂t is the infinitesimal generator in C∞−L,L. The spaces C1−L,L and C∞−L,L are dense in Lp−L,L with 1≤p<∞. The solution is represented by
(103)ut,x=Utu0,where Ut is a semigroup generated by μ−1/2∂x2 under the Dirichlet-zero boundary condition.
By changing the evolution direction fromt to x, the heat equation
(104)∂x2ut,x−μ1/2∂tut,x=0,t∈0,∞,x∈−L,L,(105)ut,−L=v0t,∂xut,−L=v1t,t∈0,∞,(106)u0,x=0x∈−L,L,is considered for x direction, where v0t and v1t are given initial functions. To establish the existence of semigroup for the x direction, it is sufficient to consider the generation of semigroup in L2−∞,∞ by generalizing t-interval from 0,∞ to −∞,∞. The Fourier transform leads to
(107)∂x2u~−iμ1/2ωu~=0u~ω,0=v~0ω,∂xu~ω,0=v~1ω,where ω is a real number. Indeed, the following transforms
(108)ut,x=12π∫−∞∞u~∫−∞∞u~ω,xeiωtdω,ut,−L=12π∫−∞∞u~ω,−Leiωtdω,∂xut,−L=12π∫−∞∞∂xu~ω,−Leiωtdω,are implemented. By solving the characteristic equation λ2−iμ1/2ω−0, the Fourier transformed solution of (105) is
(109)u~ω,x=v~0ω−i−iμ1/2ω−1/2v~1ω2e+iμ1/2ω1/2x+v~0ω+i−iμ1/2ω−1/2v~1ω2e+iμ1/2ω1/2x,where
(110)u~ω,0=v~0ω−i−iμ1/2ω1/2v~1ω2+v~0ω+i−iμ1/2ω1/2v~1ω2v~0ω,∂xu~ω,0=iμ1/2ω1/2v~0ω−i−iμ1/2ω−1/2v~1ω2−v~0ω+i−iμ1/2ω−1/2v~1ω2=v~1ω.
Meanwhile, based on the relativistic treatment, one-dimensional heat equation(111)∂x2u−μ1/2∂tu=0,is written as
(112)∂xuv−0Ifμ1/2∂t0uv=0.
Let a linear operatorA be defined by
(113)A==0Iμ1/2∂t0,in L2−∞,∞×L2−∞,∞ and the domain space of A be
(114)DA=H1−∞,∞×L2−∞,∞,where H1−∞,∞=ut∈L2−∞,∞;u′t∈L2−∞,∞,u0=0 is a Sobolev space. The Fourier transform means that the diagonalization of A is equal to
(115)A~=μ1/2∂t1/200−μ1/2∂t−1/2.
In this context, the master equation of the problem (105) is reduced to the abstract evolution equation
(116)∂xu~v~−A~u~v~=0,in L2−∞,∞×L2−∞,∞, where
(117)u~=u−i−iμ1/2ω−1/2v2,v~=u+i−iμ1/2ω−1/2v2.
It suggests that the evolution operator of equation (105) is generated by
(118)±μ1/2∂t1/2,so that it is sufficient to show ±μ1/2∂t1/2, as the infinitesimal generator. Note that the operator A~ is not necessarily a generator of analytic semigroup, because the propagation of singularity should be different if the evolution direction is different. Consequently, the existence of semigroup for (116) in the x direction is reduced to show ±μ1/2∂t1/2, as the infinitesimal generator in L2−∞,∞. In the following, the property of μ1/2∂t is discussed in the second step, and the fractional power of μ1/2∂t1/2 is studied in the third step.
Step 2. First order differential operator. The following lemma is proved in this step.
Lemma 18. The operator μ1/2∂t with the domain H1−∞,∞ is the infinitesimal generator in L2−∞,∞.
Proof of Lemma 18 Letλ be a complex number satisfying Reλ>0. First, the existence of λ−μ1/2∂t−1 is examined. Let ft be included in L2−∞,∞. Because
(119)λ−μ1/2∂tu=f,in one-dimensional interval −∞,∞ is a first-order ordinary differential equation with a constant coefficient, and the global-in-t solution necessarily exists for a given u0=u0∈ℂ. That is, λ/μ1/2 is included in the resolvent set of ∂t for an arbitrary complex number λ so that λ−μ1/2∂t−1 is concluded to be well-defined in H1−∞,∞.
Second, the resolvent operatorλ−μ1/2∂t−1 is estimated from the above. Since λ/μ1/2 is included in the resolvent set of ∂t, it is readily seen that λ−μ1/2∂t−1 is a bounded operator on L2−∞,∞. More precisely, let us consider equation (119) being equivalent to
(120)∂tut=λμ1/2ut−ftμ1/2.
If the inhomogeneous term satisfiesft∈L2−∞,∞,
(121)ut=−1μ1/2∫t∞exp−λμ1/2s−tfsds,satisfies equation (120). According to the Schwarz inequality,
(122)∫−∞∞ut2dt=∫−∞∞1μ1/2∫t∞exp−λμ1/2s−tfsds2dt≤1μ∫−∞∞∫t∞exp−Reλ2μ1/2s−texp−Reλ2μ1/2s−tfsds2dt≤1μ∫−∞∞∫t∞exp−Reλμ1/2s−tds∫t∞exp−Reλμ1/2s−tfs2dsdt,is obtained, and the equality
(123)∫t∞exp−Reλμ1/2s−tds=∫0∞exp−Reλμ1/2sds=μ1/2Reλ,is positive if Reλ>0 is satisfied. Its application leads to
(124)∫−∞∞ut2dt≤1μμ1/2Reλ∫−∞∞∫t∞exp−Reλμ1/2s−tfs2dsdt=1μμ1/2Reλ∫−∞∞∫−∞sexp−Reλμ1/2s−tdtfs2ds.
Further application of the equality(125)∫−∞sexp−Reλμ1/2s−tdt=∫−∞0expReλμ1/2tdt=μ1/2Reλ,results in
(126)∫−∞∞ut2dt≤1μμ1/2Reλ∫−∞∞μ1/2Reλfs2ds=1Reλ2∫−∞∞fs2ds,for Reλ>0, and therefore,
(127)λI−μ1/2∂t−1ftL2−∞,∞≤1ReλftL2−∞,∞,follows. That is, for Reλ>0,
(128)λI−μ1/2∂t−1≤1/Reλ,is valid. The surjective property of λI−μ1/2∂t is seen by the unique existence of solutions u∈L2−∞,∞ for the Cauchy problem of equation (120).
A semigroup is generated by taking a subset of the complex plane as(129)Ω=λ∈ℂ;λ=λ¯,where Ω is included in the resolvent set of μ1/2∂t. For λ∈Ω,λI−μ1/2∂t−1 exists, and
(130)λI−μ1/2∂t−n≤1/Reλn,is obtained. Consequently, according to the Lumer-Phillips theorem [38] for the generation of quasi contraction semigroup, μ1/2∂t is confirmed to be an infinitesimal generator in L2−∞,∞, and the unique existence of global-in-x weak solution follows [Q.E.D.: Lemma 18].
The semigroup generated byμ1/2∂t is represented by
(131)exphμ1/2∂tut=ut+μ1/2h,−∞<h<∞,so that the group is actually generated by μ1/2∂t. Indeed, the similar estimate as equation (130) can be obtained for Reλ<0 with λ−μ1/2u=f in which the solution u is represented by
(132)ut=−1μ1/2∫−∞texp−λμ1/2s−tfsds,that should be compared to equation (121).
Step 3. Fractional powers of operator. The following lemma is proved in this step.
Lemma 19.  For 0<α<1, the operator μ1/2∂tα is the infinitesimal generator in L2−∞,∞.
Proof of Lemma 19 According to Lemma 18,μ1/2∂t is the infinitesimal generator in L2−∞,∞. For an infinitesimal generator μ1/2∂t in L2−∞,∞, let the one-parameter semigroup generated by μ1/2∂t be denoted by Vx. An infinitesimal generator μ1/2∂t is a closed linear operator in L2−∞,∞. Its fractional power
(133)μ1/2∂tα,0<α<1,has been confirmed to be well-defined by S. Bochner [39] and R.S. Phillips [40] as the infinitesimal generator of semigroup (cf. K. Yosida [41]):
(134)Wxw0=∫0∞Vxw0dγλ,for w0∈L2−∞,∞, where Wx is the semigroup for x direction. The measure dγλ≥0 is defined through the Laplace integral
(135)exp−tkα=∫0∞exp−λkdγλ,where t,k>0 is satisfied [Q.E.D.: Lemma 19].
By takingα=1/2, μ1/2∂t1/2 is confirmed to be an infinitesimal generator in L2−∞,∞. Because Ω is included in the resolvent set of −μ1/2∂t1/2, it is readily seen that −μ1/2∂t1/2 is also an infinitesimal generator in L2−∞,∞.
Step 4. Abstract form of the Cole-Hopf transform. As in the original derivation of the Cole-Hopf transform, the solution of heat equation wt,x=Wx,ξwξt solved along the x direction permits its logarithm function. The abstract case of the original Cole-Hopf transform is included in the description of the logarithmic representation (100). Indeed, let an invertible evolution family Wx,ξ0≤x,ξ≤L be generated by Ax for 0≤x,ξ≤L. According to Lemma 19, the logarithmic representation of relativistic form (99) is obtained in this case as
(136)Axwξ=I+κWξ,x∂xLogWx,ξ+κIwξ,and hence as
(137)Axwx,ξwξ=κI+Wx,ξ∂xLogWx,ξ+κIwξ,using the commutation assumption. The nonlinear Anzatz −2μ−1/2∂xut,xut,x−1 of the Burgers’ equation
(138)ψt,x=−2μ−1/2∂xlogut,x=−2μ−1/2∂xut,xut,x−1,is essentially represented by
(139)−2μ−1/2AxWx,ξWx,ξ−1=−2μ−1/2κI+Wx,ξ∂xLogWx,ξ+κIWx,ξ−1=−2μ−1/2κWξ,x+I∂xLogWx,ξ+κI,in the abstract form. The similarity between equation (101) and the standard definition of operator norm is clear. In particular, the evolution direction is generalized from x to xi in equation (101).
Step 5. Generalization property. Equation (100) is the generalization of the Cole-Hopf transform. According to the introduction of nonzero κ in the abstract form, the applicability is significantly increased so that the linear equation is not necessarily the heat equation. According to the abstract nature of the logarithmic representation, linear and nonlinear equations are not necessarily considered in the one-spatial dimension. According to the relativistic treatment, the transformed variable is not limited to the spatial variables [Q.E.D.: Theorem 17].The generalized Cole-Hopf transform (100) shows that the nonlinearity of semigroup can appear simply by altering the evolution direction under a suitable identification between the infinitesimal generator and the evolution operator. In this sense, equation (90) is regarded as a local-in-xi linearized equation, if Uxi,ξi is a nonlinear semigroup (semigroup related to the nonlinear equations). Furthermore, the generalized Cole-Hopf transform (100) suggests that the relation between evolution operator and its infinitesimal generator corresponds essentially to the transform between linearity and nonlinearity. In the same context of generalizing Miura transform between the Korteweg-de-Vries and the modified Korteweg-de-Vries equations, the logarithmic representation is utilized [15].


## 5. Algebraic Structure of Infinitesimal Generators

### 5.1.BX-Module

The algebraic structure is studied based on the relativistic form of abstract equations. The operatoraxi,ξi=LogUxi,ξi+κI is bounded on X. It follows that eaxi,ξi is well-defined by the convergent power series. Note again that eaxi,ξi can be defined without assuming well-defined ∂taxi,ξi. Even without taking into account the detail property of the infinitesimal generator ∂taxi,ξi, the exponentiability is realized by the boundedness of axi,ξi. In this section, beginning with axi,ξi=LogUxi,ξi+κI, an algebraic module over a Banach algebra is defined. The essential idea of presenting a useful algebraic structure is not to examine directly the set of ∂taxi,ξi but to focus on the set of axi,ξi at first, and then, the algebraic structure of infinitesimal generators is discovered in the next. Although axi,ξi∈BX is trivially the infinitesimal generator, what is explained here is the structure of the set of preinfinitesimal generators ∂taxi,ξi.Theorem 20.
(normed vector space). LetUjxi,ξi be evolution operators satisfying equation (19) and LogUjxi,ξi be well-defined for any xi,ξi∈−L,L and j=1,2,⋯,n. Log Ujxi,ξi is assumed to commute with each other.
(140)VLgX≔kLogUjxi,ξi;k∈C,xi,ξi∈−L,L⊂BX⊂GX,is a normed vector space over the complex number field, where BX denotes a set of all the bounded operators on X.
Proof. In case of κ=0, the operator axi,ξi is reduced to
(141)LogUxi,ξi∈BX.
The operator sum is calculated using the Dunford-Riesz integral(142)LogUxi,ηi+LogUηi,ξi=12πi∫ΓLogλλI−Uxi,ηi−1dλ+12πi∫Γ′Logλ′λ′I−Uηi,ξi−1dλ′=12πi2∫Γ∫Γ′Logλ+Logλ′λI−Uxi,ηi−1λ′I−Uηi,ξi−1dλ′dλ=12πi2∫Γ∫Γ′Logλλ′λI−Uxi,ηi−1λ′I−Uηi,ξi−1dλ′dλ=LogUxi,ηiUηi,ξi=LogUxi,ξi,then, the sum closedness is clear. Here, Γ′ is assumed to be included in Γ, and this condition is not so restrictive in the present setting. In a different situation, when Ut,r and Vt,r commute for the same t and r, another kind of sum is calculated as
(143)LogU1xi,ηi+LogU2xi,ηi=12πi∫ΓLogλλI−U1xi,ηi−1dλ+12πi∫Γ′Logλ′λ′I−U2xi,ηi−1dλ′=12πi2∫Γ∫Γ′Logλ+Logλ′λI−U1xi,ηi−1λ′I−U2xi,ηi−1dλ′dλ=12πi2∫Γ∫Γ′Logλλ′λI−U1xi,ηi−1λ′I−U2xi,ηi−1dλ′dλ=LogU1xi,ηiU2xi,ηi,where for Wxi,ηi=eLogU1xi,ηi+LogU2xi,ηi=U1xi,ηiU2xi,ηi, the semigroup property is satisfied as
(144)Wxi,ηiWηi,ξi=U1xi,ηiU2xi,ηiU1ηi,ξiU2ηi,ξi=U1xi,ηiU1ηi,ξiU2xi,ηiU2ηi,ξi=Wxi,ξi,Wξi,ξi=Wξi,xiWxi,ξi=I,and then, the sum closedness is clear. Although the logarithm function is inherently a multivalued function, the uniqueness of sum operation is ensured by the single-valued property of the principal branch “Log.” Consequently, since the closedness for scalar product is obvious, where
(145)W~xi,ηi=ek1LogU1xi,ηi+k2LogU2xi,ηi=ek1LogU1xi,ηiek2LogU2xi,ηi,holds the semigroup property in a similar way to Wxi,ηi, Consequently, VLgX is a normed vector space over the complex number field. In particular, the zero operator LogI is included in VLgX. Theorem 1 has been proved [Q.E.D.: Theorem 20].Theorem 21.
(BX-module). Let Ujxi,ξi be evolution operators satisfying equation (19) for any xi,ξi∈−L,L and j=1,2,⋯,n. For a certain K∈BX, let a subset of BX in which each element is assumed to commute with LogUjxi,ξi+K are assumed to commute with each other.
(146)BLgX≔KLogUjxi,ξi+K;K∈BabX,K∈BX,x,ξ∈−L,L⊂BX⊂GX,is a module over the Banach algebra.
Proof. It is worth generalizing the above normed vector space. In this sense, utilizing a common operator K∈BX, components are changed to LogUxi,ξi+K.
The operator sum is calculated as(147)LogUxi,ηi+K+LogUηi,ξi+K=12πi∫ΓLogλλI−Uxi,ηi−K−1dλ+12πi∫Γ′Logλ′λ′I−Uηi,ξi−K−1dλ′=12πi2∫Γ∫Γ′Logλ+Logλ′λI−Uxi,ηi−K−1λ′I−Uηi,ξi−K−1dλ′dλ=12πi2∫Γ∫Γ′Logλλ′λI−Uxi,ηi−K−1λ′I−Uηi,ξi−K−1dλ′dλ=LogUxi,ηi+KUηi,ξi+K=LogUxi,ξi+KUxi,ηi+KUηi,ξi+K2.
After introducing a certainK∈BX with sufficient large ∥K∥, it is always possible to take integral path Γ′ to be included in Γ. Since the part “KUt,r+KUr,s+K2I” is included in BX, the sum-closedness is clear. In a different situation, when U1t,r and U2t,r commute for the same t and r, another kind of sum is calculated as
(148)LogU1xi,ηi+K1+LogU2xi,ηi+K2=12πi∫ΓLogλλI−U1xi,ηi−K1−1dλ+12πi∫Γ′Logλ′λ′I−U2xi,ηi−K2−1dλ=12πi2∫Γ∫Γ′Logλ+Logλ′λI−U1xi,ηi−K1−1λ′I−U2xi,ηi−K2−1dλ′dλ=12πi2∫Γ∫Γ′Logλλ′λI−U1xi,ηi−K1−1λ′I−U2xi,ηi−K2−1dλ′dλ=LogU1xi,ηi+K1U2xi,ηi+K2=LogWxi,ηi+K2U1xi,ηi+K1U2xi,ηi+K1K2.
Since the part “K2U1xi,ηi+K1U2xi,ηi+K1K2” is included in BX, the sum-closedness is clear.
The productKLogU1xi,ξi+K2∈BX is justified by the operator product equipped with BX. Since the closedness for operator product within BX is obvious, where
(149)W^xi,ηi=eK1LogU1xi,ηi+K2LogU2xi,ηi=eK1LogU1xi,ηieK2LogU2xi,ηi,holds the semigroup property, and using an identity operator I∈BX,
(150)KLogUxi,ηi+K−KLogUxi,ηi=KLogI+KUxi,ηi−1,and therefore
(151)K1LogU1xi,ηi+K1+K2LogU2xi,ηi+K2=LogU^xi,ηi+I+LogW^xi,ηi,=LogW^xi,ηi+U^xi,ηiW^xi,ηi,are valid for U^xi,ηi≔eK1LogI+K1U1xi,ηi−1+K2LogI+K2U2xi,ηi−1−I. Consequently, BLgX is a module over a Banach algebra. In particular, a relation VLgX⊂BLgX is satisfied. The statement has been proved [Q.E.D.: Theorem 21].The next corollary follows.Corollary 22.
(BX-module for infinitesimal generators). Let Ujxi,ξi be evolution operators satisfying equation (19) for any xi,ξi∈−L,L and j=1,2,⋯,n. For a certain K∈BX, let a subset of BX in which each element is assumed to commute with LogUjxi,ξi+K be BabX. LogUjxi,ξi+K are assumed to commute with each other.
(152)GLgX≔K∂xiLogUjxi,ξi+K;K∈BabX,K∈BabXx,ξ∈−L,L⊂GX,is a module over a Banach algebra.
Proof. According to the linearity of differential operator, the introduction of differential operator ∂xi is true without any additional treatment. It is sufficient to see that there exists a certain ηi such that
(153)K∂xiLogUjxi,ξi+K=∂xi∫ξixiK∂ηiLogUjηi,ξi+Kdηi∂xiKLogUjxi,ξi+K−KLogI+K−∫ξixi∂ηiKdηiLogUjxi,ξi+K,according to the mean value theorem ξi≤ηi≤xi [Q.E.D.: Corollary 22].The module over a Banach algebra is calledBX-module. For the structure of BLgX, a certain originally unbounded part can be classified to LogUxi,ξi+K∈BX and the rest part to K∈BabX. Here, the terminology “originally unbounded” is used, because some unbounded operators are reduced to bounded operators under the validity of the logarithmic representation.It is necessary to connect the concept ofBX-module to the set of infinitesimal generators. Let us move on to the operator ∂xiKLogUjxi,ξi+K, which is expected to be the preinfinitesimal generator of expKLogUjxi,ξi+K. This property is surely true by the inclusion relation BLgX⊂BX. It is also suggested by the inclusion relation BLgX⊂BX, operators ∂xiLogUjxi,ξi+K are the preinfinitesimal generators if LogUjxi,ξi+K∈GLgX is satisfied. Consequently, the unbounded sum-perturbation for infinitesimal generators is seen by the sum closedness of BX-module. Note that it does not require the self-adjointness of the operator.The preinfinitesimal generator property is examined for products of operators in the next two theorems.Theorem 23.
(product perturbation for preinfinitesimal generators). For a certainK∈BX, let a subset of BX in which each element is assumed to commute with LogUixi,ξi+K be BabX. Let an operator denoted by
(154)LLogUxi,ξi+K,be included in BabX, where the evolution operator Ut,s on X is generated by At, L is an element in BabX, and K is an element in BX. Let K and L be further assumed to be independent of xi. The product of preinfinitesimal generators, which is represented by
(155)LAt=I−Ke−axi,ξi−1∂xiLLogUxi,ξi+K,is also the preinfinitesimal generators in X, where axi,ξi=LogUxi,ξi+K.
Proof. Since L is independent of xi,
(156)∂xiLLogUxi,ξi+K=L∂xiLogUxi,ξi+K,is true. The basic calculi using the t-independence of K leads to the product of operator LAt. It is well-defined by
(157)I−Ke−axi,ξi−1∂xiLLogUxi,ξi+K=LI−Ke−axi,ξi−1∂xiLogUxi,ξi+K=LI−Ke−axi,ξi−1Uxi,ξi+K−1∂xiUxi,ξi=LAt,under the commutation assumptions, where the relation ∂xiUxi,ξi=AxiUxi,ξi is applied. Let xi,ξi∈−L,+L satisfy ξi<xi. The preinfinitesimal generator property of LAt is confirmed by
(158)∫ξixiI−Ke−axi,ξi−1∂ηiLLogUηi,ξi+Kdηi≤I−Ke−axi,ξi−1∫ξixi∂ηiLLogUηi,ξi+Kdηi≤supηi∈xi,ξiI−Ke−axi,ξi−1∫ξixi∂ηiLLogUηi,ξi+Kdηi≤supηi∈xi,ξiI−Ke−axi,ξi−1LLogUxi,ξi+K−LogI+K,where a certain real number σ∈ξi,xi is determined by the mean value theorem. Consequently, due to the boundedness of ∫ξixiI−Ke−axi,ξi−1∂xiLLogUxi,ξi+Kdηi on X, I−Ke−axi,ξi∂xiLLogUxi,ξi+K is confirmed to be the preinfinitesimal generator in X [Q.E.D.: Theorem 23].The operatorL can be regarded as a perturbation to the operators in BLgX. This lemma shows the product perturbation for the infinitesimal generators of C0-semigroups under the commutation, although the perturbation has been studied mainly for the sum of operators. It is remarkable that the self-adjointness of the operator is not required for this lemma. For the details of conventional bounded sum perturbation and the perturbation theory for the self-adjoint operators, see Ref. [42].Theorem 24.
(operator product). For a certainKxi∈BX, let a subset of BX in which each element is assumed to commute with LogUxi,ξi+Kxi be BabX. Let an operator denoted by
(159)LLogUxi,ξi+Kxi,be included in BLgX, where the evolution operator Uxi,ξi on X is generated by Axi, L is an element in BabX, and Kxi is an element in BX. Let L and Kxi be xi-independent and xi-dependent, respectively. The operators represented by
(160)Lxi∂xiLLogUxi,ξi+Kxi,is the preinfinitesimal generators in X, if the operator Lηi∈BX is strongly continuous with respect to ηi in the interval ξi,xi.
Proof. Let xi,ξi∈−L,+L satisfy ξi<xi. The preinfinitesimal generator property is reduced to the possibility of applying the mean value theorem.
(161)∫ξixiLηi∂ηiLogUηi,ξi+Kηidηi≤Lσi∫ξixi∂ηiLogUηi,ξi+Kηidηi≤supσi∈xi,ξiLσi∫ξixi∂ηiLogUηi,ξi+Kηidηi≤supσi∈xi,ξiLσiLogUηi,ξi+Kxi−LogI+Kxi,where a certain real number σi∈ξi,xi is determined by the mean value theorem. Consequently, Lxi∂xiLLogUxi,ξi+Kxi is confirmed to be the preinfinitesimal generator in X [Q.E.D.: Theorem 24].Equation (160) provides one standard form for the representation of operator products in the sense of logarithmic representation. Consequently, BX-module is associated with the preinfinitesimal generator.


### 5.2. Formulation of Rotation Group

The application example ofBX-module is provided. The concept of BX-module is generally enough to provide a foundation of the conventional bounded formulation of Lie algebras (for a textbook, see [43]). In other words, by means of BX-module, the intersection of the Banach algebra (including only bounded operators) and the extracted bounded part of the Lie algebra (generally including unbounded operators) is shown. More precisely, using BX-module, the bounded part is extracted from unbounded angular momentum operators. The extracted bounded parts are utilized to formulate the rotation group with incorporating the unboundedness of angular momentum algebra.The mathematical foundation of rotation group is demonstrated [14]. Although the evolution parameter in this paper is denoted by t,s∈−T,+T, it is more likely to be denoted by θ,σ∈−Θ,+Θ, because the evolution parameter in the present case means the rotation angle. The rotation group is generated by the angular momentum operator (for textbooks, see Refs. [44, 45]). The angular momentum operator includes a differential operator, as represented by
(162)L−ihr×∇,where ℏ is a real constant called the Dirac constant. The appearance of differential operator ∇ in the representation of L is essential. The operator ∇ is an unbounded operator for example in a Hilbert space L2R3, while it must be treated as a bounded operator in terms of establishing an algebraic ring structure. Furthermore, the operator boundedness is also indispensable for some important formulae such as the Baker-Campbell-Hausdorff formula and the Zassenhaus formula to be valid. In general, the exponential of unbounded operators cannot be represented by the power series expansion (cf. the Yosida approximation in a typical proof of the Hille-Yosida theorem; e.g., see Ref. [22]).Letℝ3 be the three-dimensional spatial coordinate spanned by the standard orthogonal axes, x, y, and z. The angular momentum operator L is considered in L2ℝ3. The angular momentum operator
(163)L=Lx,Ly,Lz,consists of x, y, and z components
(164)Lx=ihy∂z−z∂y,Ly=ihz∂x−x∂z,Lz=ihy∂y−z∂x,respectively. The commutation relations
(165)Lx,Ly=ihLz,Ly,Lz,ihLx,Lz,Lx=ihLy,are true, where Li,Lj≔LiLj−LjLi denotes a commutator product i,j=x,y,z. The commutation of angular momentum operators arises from the commutation relations of the canonical quantization
(166)x,px=y,py=z,pz=ih,y,px=y,pz=z,py=x,py=x,pz=0.Indeed, the momentum operatorp=px,py,pz is represented by p=ih∂x,∂y,∂z in quantum mechanics. It is remarkable that the commutation is always true for the Newtonian mechanics; i.e., x,px=y,py=z,pz=0 is true in addition to y,px=y,pz=z,px=z,py=x,py=x,pz=0.Let a set of all bounded operators onL2ℝ3 be denoted by BL2ℝ3. A set of operators Lk;k=x,y,z or iLk/h;k=x,y,z with the commutation relation (165) is regarded as the Lie algebra. In particular α^Lx+β^Ly+γ^Lz;α^,β^,γ^∈ℂ forms a vector space over the complex number field, while αiLx/h+βiLy/h+γiLz/h;α,β,γ∈ℝ is a vector space over the real number field. It is possible to associate the real numbers α, β, and γ with the Euler angles (for example, see Ref. [46]). The second term of the right hand side of
(167)ri∂jrk∂l=ri∂jrk∂l+rirk∂j∂l,disappears as far as the commutator product Li,Lj is concerned, where ri is equal to i, and i,j,k,l=x,y, or z satisfy i≠j and k≠l. This fact is a key to justify the algebraic ring structure of Lk;k=x,y,z. On the other hand, although Lk is assumed to be bounded on L2ℝ3 in the typical treatment of the Lie algebra, it is not the case for the angular momentum algebra because of the appearance of differential operators in their definitions. From a geometric point of view, the range space RLk⊂L2ℝ3 strictly includes the domain space DLk; i.e., there is no guarantee for any u∈L2ℝ3 and a certain positive M∈ℝ to satisfy LkuL2ℝ3≤MuL2ℝ3. In order to establish iLk/h;k=x,y,z as the Lie algebra, it is necessary to show
(168)±iLk/h=±ri∂rj−rj∂ri,as an infinitesimal generator in Lr2ℝ3, where i,j,k=x,y,z satisfies i≠j≠k. As for the angular momentum operator, the t-independent assumption for operators K and L in Theorem 23 and Theorem 24 is satisfied. Note that t-independence assumes a kind of commutation relation.Theorem 25.
(unbounded formulation of rotation group). Letri be either x,y, or z. For i≠j, an operator ±ri∂rj with its domain space Hr2ℝ3 is an infinitesimal generator in Lr2ℝ3. Consequently, the angular momentum operators
(169)±iLx/h=±y∂z−z∂y,±iLy/h=±z∂x−x∂z,±iLz/h=±x∂y−y∂x,are infinitesimal generators in Lr2ℝ3.
Proof. The proof consists of three steps.
Step 1. ∂rk as an infinitesimal generator.
Lemma 26. For rk equal to x,y, or z, an operator ∂rk with its domain H1ℝ3 is an infinitesimal generator in Lr2ℝ3.
Proof of Lemma 26 The operator∂x is known as the infinitesimal generator of the first order hyperbolic type partial differential equations. For a complex number λ satisfying Reλ>0, let us consider a differential equation
(170)∂xux=λux−fx,in L2ℝ, and
(171)ux=−∫x∞expλx,ξfξdξ,satisfies the equation. According to the Schwarz inequality,
(172)∫−∞+∞ux2dx=∫−∞+∞∫x∞expλx,ξfξdξ2dx≤∫ξixi∫x∞expReλx−ξ2expReλx−ξ2fξdξ2dx≤∫ξixi∫x∞expReλx−ξdξ∫x∞expReλx−ξfξ2dξdx,is obtained, because eλ/22=eReλ/22eiImλ/22≤eReλ is valid if Reλ>0. Here, the equality
(173)∫x∞expReλx−ξdξ=∫0∞exp−Reλξdξ=1Reλ,is positive valued if Reλ>0. Its application leads to
(174)∫−∞+∞ux2dx≤1Reλ∫−∞+∞∫x∞expReλx−ξfξ2dξdx≤1Reλ∫−∞+∞∫−∞ξexpReλx−ξdxfξ2dξ.
Further application of the equality(175)∫−∞ξexpReλx−ξdx=∫−∞0expReλxdx=1Reλ,results in
(176)∫−∞+∞ux2dx≤1Reλ2∫−∞+∞fξ2dξ,and therefore
(177)λI−∂x−1fL2ℝ≤1Reλ2fL2ℝ.
That is, forReλ>0,
(178)λI−∂x−1≤1Reλ,is valid. The surjective property of λI−∂k is seen by the unique existence of solution ux∈L2ℝ for the initial value problem of equation (170).
A semigroup is generated by taking a subset of the complex plane as(179)Ω=λ∈ℂ,λ=λ¯,where Ω is included in the resolvent set of ∂x. For λ∈Ω,λI−∂x−1 exists, and
(180)λI−∂x−n≤1Reλn,is obtained. Consequently, according to the Lumer-Phillips theorem [38, 40] for the generation of quasi contraction semigroup, ∂x with the domain space H1ℝ3 is confirmed to be an infinitesimal generator in L2ℝ. The similar argument is valid to ∂y and ∂z. By considering x,y,z∈ℝ3, ∂k with k=x,y,z is the infinitesimal generators in L2ℝ3 [Q.E.D.: Lemma 26].
Step 2. irkI as an infinitesimal generator.
Lemma 27. Let rk be either x,y, or z. Let I be the identity operator of L2ℝ. An operator irkI is an infinitesimal generator in L2ℝ.
Proof of Lemma 27 For anyw∈ℂ, it is possible to define the exponential function by the convergent power series: ew=∑j=0∞wj/j!, so that
(181)eitrkI=∑j=0∞1j!itrkIj,is well-defined for t,rk∈ℝ. This fact is ensured by the boundedness of the identity operator I, although rkI and irkI are not bounded operators in L2ℝ if the standard L2-norm is equipped. It is sufficient for irkI to be the preinfinitesimal generator.
For an arbitraryrk∈ℝ, an operator irkI with its domain L2ℝ is the infinitesimal generator in L2ℝ; indeed, the spectral set is on the imaginary axis of the complex plane, and the unitary operator is generated as
(182)∫eitrkIu2drk=∫u2drk.
Consequently, the operatorirkI is treated as an infinitesimal generator in L2ℝ and therefore in L2ℝ3 [Q.E.D.: Lemma 27].
Step 3. ±ri∂rj−rj∂ri as an infinitesimal generator. Let i≠j be satisfied for i,j=x,y,z. Since r±t−s∂rj is well-defined (cf. Lemma 2) with the domain space Hr1ℝ3, its logarithmic representation is obtained by
(183)±iri∂rj=iriI−κe±s−t∂rj∂tLoge±t−s∂rj+κI,where κ≠0 is a certain complex number. According to Theorem 24, the product between iriI and ±∂rj is represented by
(184)±iri∂rj=iriI−κe±s−t∂rj∂tLoge±t−s∂rj+κI.
Using the commutation andt-independence of riI, it leads to the logarithmic representation
(185)±ri∂rj=I−κe±s−t∂rj∂triLoge±t−s∂rj+κI,without the loss of generality. The domain space of riI is equal to L2ℝ3, as eiriI is represented by the convergent power series in Lr2ℝ3. The half plane λ∈ℂ;Reλ>0 is included in the resolvent set of ±ri∂rj. Consequently, for t,ri∈ℝ, the existence of e±tri∂rj directly follows from the confirmed existence of e±t∂rj. Being equipped with the domain space Hr1ℝ3,±ri∂rj is the infinitesimal generator in Lr2ℝ3.
The preinfinitesimal generator property of sum is also understood by theBX-module property. The sum between ri∂ri and −rj∂ri is represented by
(186)I+κe+s−t∂rj∂triLoge+t−s∂rj+κI−I+κe−s−t∂ri∂trjLoge−t−s∂ri+κI=I+κe+s−t∂rj∂triLoge+s−t∂rj+κI−rjLoge−t−s∂ri+κI−κe+s−t∂rj−κe−s−t∂ri∂trjLoge−t−s∂ri+κI=I+κe+s−t∂rj∂triLoge+t−s∂rj+κI−Loge−t−s∂ri+κI−I+κe+s−t∂ri∂tri−rjLoge−t−s∂ri+κI−κe+s−t∂rj−κe−s−t∂ri∂trjLoge−t−s∂ri+κI,where all the three terms in the right hand side are of the form
(187)Lt∂triLogUt,s+Kt,whose preinfinitesimal generator properties are proved similarly to Lemma 26. In particular, the first term in the right hand side can be reduced to the above form with L=1 and t-dependent Kt, the parts corresponding to Lt are strongly continuous, and ri is independent of t. After having an integral of equation (186) in terms of t, each term is regraded as a bounded operator on Lr2ℝ3. Consequently, for i≠j, the application of Lemma 26 leads to the fact that
(188)±ri∂rj−rj∂ri,with its domain space Hr1ℝ3 is the infinitesimal generator in Lr2ℝ3 [Q.E.D.: Theorem 25].Corollary 28.
(collective renormalization). Fort,s∈−T,+T, let Vkt,s with k=x,y,z in Lr2ℝ3 be generated by iLk/h. For a certain complex constant ≠0, the angular momentum operator ±iLk/h with k=x,y,z is represented by the logarithm
(189)±iLk/h=±I+κVks,t∂tLogVkt,s+κI,and the corresponding evolution operator is expanded by the convergent power series
(190)Vkt,s=eLogVkt,s+κI−κI=∑n−0∞1n!LogVkt,s+κIn−κI=1−κI+∑n−0∞1n!LogVkt,s+κIn,where LogVkt,s+κI is bounded on Lr2ℝ3, although Lk is unbounded in Lr2ℝ3.
Proof. The group Vkt,s with k=x,y,z is generated by the infinitesimal generator iLk/h in Lr2ℝ3. This fact leads to the logarithmic representation
(191)iLk/h=I+κVks−t∂tLogVkt,s+κI,where κ≠0 is a certain complex constant. The relation Vkt,s+κI=eLogVkt,s+κI admits the power series expansion of Vkt,s [Q.E.D.: Corollary 28].Equation (189) shows a convergent power series representation for the rotation group. Let us call the representation shown in equation (189) the collective renormalization (cf. renormalized evolution equation in Corollary 9), in which a detailed degree of freedom ri∂rj is switched to a collective degree of freedom Lk. According to the collective renormalization, the evolution problem is studied by beginning with the bounded evolution operator Vks,t and the related bounded infinitesimal generator LogVkt,s+κI. In a more mathematical sense, the collective renormalization plays a role of simplifying the representation. Equation (190) ensures the validity of convergent power series expansions used in operator algebras even if they include unbounded operators.


## 5.1.BX-Module

The algebraic structure is studied based on the relativistic form of abstract equations. The operatoraxi,ξi=LogUxi,ξi+κI is bounded on X. It follows that eaxi,ξi is well-defined by the convergent power series. Note again that eaxi,ξi can be defined without assuming well-defined ∂taxi,ξi. Even without taking into account the detail property of the infinitesimal generator ∂taxi,ξi, the exponentiability is realized by the boundedness of axi,ξi. In this section, beginning with axi,ξi=LogUxi,ξi+κI, an algebraic module over a Banach algebra is defined. The essential idea of presenting a useful algebraic structure is not to examine directly the set of ∂taxi,ξi but to focus on the set of axi,ξi at first, and then, the algebraic structure of infinitesimal generators is discovered in the next. Although axi,ξi∈BX is trivially the infinitesimal generator, what is explained here is the structure of the set of preinfinitesimal generators ∂taxi,ξi.Theorem 20.
(normed vector space). LetUjxi,ξi be evolution operators satisfying equation (19) and LogUjxi,ξi be well-defined for any xi,ξi∈−L,L and j=1,2,⋯,n. Log Ujxi,ξi is assumed to commute with each other.
(140)VLgX≔kLogUjxi,ξi;k∈C,xi,ξi∈−L,L⊂BX⊂GX,is a normed vector space over the complex number field, where BX denotes a set of all the bounded operators on X.
Proof. In case of κ=0, the operator axi,ξi is reduced to
(141)LogUxi,ξi∈BX.
The operator sum is calculated using the Dunford-Riesz integral(142)LogUxi,ηi+LogUηi,ξi=12πi∫ΓLogλλI−Uxi,ηi−1dλ+12πi∫Γ′Logλ′λ′I−Uηi,ξi−1dλ′=12πi2∫Γ∫Γ′Logλ+Logλ′λI−Uxi,ηi−1λ′I−Uηi,ξi−1dλ′dλ=12πi2∫Γ∫Γ′Logλλ′λI−Uxi,ηi−1λ′I−Uηi,ξi−1dλ′dλ=LogUxi,ηiUηi,ξi=LogUxi,ξi,then, the sum closedness is clear. Here, Γ′ is assumed to be included in Γ, and this condition is not so restrictive in the present setting. In a different situation, when Ut,r and Vt,r commute for the same t and r, another kind of sum is calculated as
(143)LogU1xi,ηi+LogU2xi,ηi=12πi∫ΓLogλλI−U1xi,ηi−1dλ+12πi∫Γ′Logλ′λ′I−U2xi,ηi−1dλ′=12πi2∫Γ∫Γ′Logλ+Logλ′λI−U1xi,ηi−1λ′I−U2xi,ηi−1dλ′dλ=12πi2∫Γ∫Γ′Logλλ′λI−U1xi,ηi−1λ′I−U2xi,ηi−1dλ′dλ=LogU1xi,ηiU2xi,ηi,where for Wxi,ηi=eLogU1xi,ηi+LogU2xi,ηi=U1xi,ηiU2xi,ηi, the semigroup property is satisfied as
(144)Wxi,ηiWηi,ξi=U1xi,ηiU2xi,ηiU1ηi,ξiU2ηi,ξi=U1xi,ηiU1ηi,ξiU2xi,ηiU2ηi,ξi=Wxi,ξi,Wξi,ξi=Wξi,xiWxi,ξi=I,and then, the sum closedness is clear. Although the logarithm function is inherently a multivalued function, the uniqueness of sum operation is ensured by the single-valued property of the principal branch “Log.” Consequently, since the closedness for scalar product is obvious, where
(145)W~xi,ηi=ek1LogU1xi,ηi+k2LogU2xi,ηi=ek1LogU1xi,ηiek2LogU2xi,ηi,holds the semigroup property in a similar way to Wxi,ηi, Consequently, VLgX is a normed vector space over the complex number field. In particular, the zero operator LogI is included in VLgX. Theorem 1 has been proved [Q.E.D.: Theorem 20].Theorem 21.
(BX-module). Let Ujxi,ξi be evolution operators satisfying equation (19) for any xi,ξi∈−L,L and j=1,2,⋯,n. For a certain K∈BX, let a subset of BX in which each element is assumed to commute with LogUjxi,ξi+K are assumed to commute with each other.
(146)BLgX≔KLogUjxi,ξi+K;K∈BabX,K∈BX,x,ξ∈−L,L⊂BX⊂GX,is a module over the Banach algebra.
Proof. It is worth generalizing the above normed vector space. In this sense, utilizing a common operator K∈BX, components are changed to LogUxi,ξi+K.
The operator sum is calculated as(147)LogUxi,ηi+K+LogUηi,ξi+K=12πi∫ΓLogλλI−Uxi,ηi−K−1dλ+12πi∫Γ′Logλ′λ′I−Uηi,ξi−K−1dλ′=12πi2∫Γ∫Γ′Logλ+Logλ′λI−Uxi,ηi−K−1λ′I−Uηi,ξi−K−1dλ′dλ=12πi2∫Γ∫Γ′Logλλ′λI−Uxi,ηi−K−1λ′I−Uηi,ξi−K−1dλ′dλ=LogUxi,ηi+KUηi,ξi+K=LogUxi,ξi+KUxi,ηi+KUηi,ξi+K2.
After introducing a certainK∈BX with sufficient large ∥K∥, it is always possible to take integral path Γ′ to be included in Γ. Since the part “KUt,r+KUr,s+K2I” is included in BX, the sum-closedness is clear. In a different situation, when U1t,r and U2t,r commute for the same t and r, another kind of sum is calculated as
(148)LogU1xi,ηi+K1+LogU2xi,ηi+K2=12πi∫ΓLogλλI−U1xi,ηi−K1−1dλ+12πi∫Γ′Logλ′λ′I−U2xi,ηi−K2−1dλ=12πi2∫Γ∫Γ′Logλ+Logλ′λI−U1xi,ηi−K1−1λ′I−U2xi,ηi−K2−1dλ′dλ=12πi2∫Γ∫Γ′Logλλ′λI−U1xi,ηi−K1−1λ′I−U2xi,ηi−K2−1dλ′dλ=LogU1xi,ηi+K1U2xi,ηi+K2=LogWxi,ηi+K2U1xi,ηi+K1U2xi,ηi+K1K2.
Since the part “K2U1xi,ηi+K1U2xi,ηi+K1K2” is included in BX, the sum-closedness is clear.
The productKLogU1xi,ξi+K2∈BX is justified by the operator product equipped with BX. Since the closedness for operator product within BX is obvious, where
(149)W^xi,ηi=eK1LogU1xi,ηi+K2LogU2xi,ηi=eK1LogU1xi,ηieK2LogU2xi,ηi,holds the semigroup property, and using an identity operator I∈BX,
(150)KLogUxi,ηi+K−KLogUxi,ηi=KLogI+KUxi,ηi−1,and therefore
(151)K1LogU1xi,ηi+K1+K2LogU2xi,ηi+K2=LogU^xi,ηi+I+LogW^xi,ηi,=LogW^xi,ηi+U^xi,ηiW^xi,ηi,are valid for U^xi,ηi≔eK1LogI+K1U1xi,ηi−1+K2LogI+K2U2xi,ηi−1−I. Consequently, BLgX is a module over a Banach algebra. In particular, a relation VLgX⊂BLgX is satisfied. The statement has been proved [Q.E.D.: Theorem 21].The next corollary follows.Corollary 22.
(BX-module for infinitesimal generators). Let Ujxi,ξi be evolution operators satisfying equation (19) for any xi,ξi∈−L,L and j=1,2,⋯,n. For a certain K∈BX, let a subset of BX in which each element is assumed to commute with LogUjxi,ξi+K be BabX. LogUjxi,ξi+K are assumed to commute with each other.
(152)GLgX≔K∂xiLogUjxi,ξi+K;K∈BabX,K∈BabXx,ξ∈−L,L⊂GX,is a module over a Banach algebra.
Proof. According to the linearity of differential operator, the introduction of differential operator ∂xi is true without any additional treatment. It is sufficient to see that there exists a certain ηi such that
(153)K∂xiLogUjxi,ξi+K=∂xi∫ξixiK∂ηiLogUjηi,ξi+Kdηi∂xiKLogUjxi,ξi+K−KLogI+K−∫ξixi∂ηiKdηiLogUjxi,ξi+K,according to the mean value theorem ξi≤ηi≤xi [Q.E.D.: Corollary 22].The module over a Banach algebra is calledBX-module. For the structure of BLgX, a certain originally unbounded part can be classified to LogUxi,ξi+K∈BX and the rest part to K∈BabX. Here, the terminology “originally unbounded” is used, because some unbounded operators are reduced to bounded operators under the validity of the logarithmic representation.It is necessary to connect the concept ofBX-module to the set of infinitesimal generators. Let us move on to the operator ∂xiKLogUjxi,ξi+K, which is expected to be the preinfinitesimal generator of expKLogUjxi,ξi+K. This property is surely true by the inclusion relation BLgX⊂BX. It is also suggested by the inclusion relation BLgX⊂BX, operators ∂xiLogUjxi,ξi+K are the preinfinitesimal generators if LogUjxi,ξi+K∈GLgX is satisfied. Consequently, the unbounded sum-perturbation for infinitesimal generators is seen by the sum closedness of BX-module. Note that it does not require the self-adjointness of the operator.The preinfinitesimal generator property is examined for products of operators in the next two theorems.Theorem 23.
(product perturbation for preinfinitesimal generators). For a certainK∈BX, let a subset of BX in which each element is assumed to commute with LogUixi,ξi+K be BabX. Let an operator denoted by
(154)LLogUxi,ξi+K,be included in BabX, where the evolution operator Ut,s on X is generated by At, L is an element in BabX, and K is an element in BX. Let K and L be further assumed to be independent of xi. The product of preinfinitesimal generators, which is represented by
(155)LAt=I−Ke−axi,ξi−1∂xiLLogUxi,ξi+K,is also the preinfinitesimal generators in X, where axi,ξi=LogUxi,ξi+K.
Proof. Since L is independent of xi,
(156)∂xiLLogUxi,ξi+K=L∂xiLogUxi,ξi+K,is true. The basic calculi using the t-independence of K leads to the product of operator LAt. It is well-defined by
(157)I−Ke−axi,ξi−1∂xiLLogUxi,ξi+K=LI−Ke−axi,ξi−1∂xiLogUxi,ξi+K=LI−Ke−axi,ξi−1Uxi,ξi+K−1∂xiUxi,ξi=LAt,under the commutation assumptions, where the relation ∂xiUxi,ξi=AxiUxi,ξi is applied. Let xi,ξi∈−L,+L satisfy ξi<xi. The preinfinitesimal generator property of LAt is confirmed by
(158)∫ξixiI−Ke−axi,ξi−1∂ηiLLogUηi,ξi+Kdηi≤I−Ke−axi,ξi−1∫ξixi∂ηiLLogUηi,ξi+Kdηi≤supηi∈xi,ξiI−Ke−axi,ξi−1∫ξixi∂ηiLLogUηi,ξi+Kdηi≤supηi∈xi,ξiI−Ke−axi,ξi−1LLogUxi,ξi+K−LogI+K,where a certain real number σ∈ξi,xi is determined by the mean value theorem. Consequently, due to the boundedness of ∫ξixiI−Ke−axi,ξi−1∂xiLLogUxi,ξi+Kdηi on X, I−Ke−axi,ξi∂xiLLogUxi,ξi+K is confirmed to be the preinfinitesimal generator in X [Q.E.D.: Theorem 23].The operatorL can be regarded as a perturbation to the operators in BLgX. This lemma shows the product perturbation for the infinitesimal generators of C0-semigroups under the commutation, although the perturbation has been studied mainly for the sum of operators. It is remarkable that the self-adjointness of the operator is not required for this lemma. For the details of conventional bounded sum perturbation and the perturbation theory for the self-adjoint operators, see Ref. [42].Theorem 24.
(operator product). For a certainKxi∈BX, let a subset of BX in which each element is assumed to commute with LogUxi,ξi+Kxi be BabX. Let an operator denoted by
(159)LLogUxi,ξi+Kxi,be included in BLgX, where the evolution operator Uxi,ξi on X is generated by Axi, L is an element in BabX, and Kxi is an element in BX. Let L and Kxi be xi-independent and xi-dependent, respectively. The operators represented by
(160)Lxi∂xiLLogUxi,ξi+Kxi,is the preinfinitesimal generators in X, if the operator Lηi∈BX is strongly continuous with respect to ηi in the interval ξi,xi.
Proof. Let xi,ξi∈−L,+L satisfy ξi<xi. The preinfinitesimal generator property is reduced to the possibility of applying the mean value theorem.
(161)∫ξixiLηi∂ηiLogUηi,ξi+Kηidηi≤Lσi∫ξixi∂ηiLogUηi,ξi+Kηidηi≤supσi∈xi,ξiLσi∫ξixi∂ηiLogUηi,ξi+Kηidηi≤supσi∈xi,ξiLσiLogUηi,ξi+Kxi−LogI+Kxi,where a certain real number σi∈ξi,xi is determined by the mean value theorem. Consequently, Lxi∂xiLLogUxi,ξi+Kxi is confirmed to be the preinfinitesimal generator in X [Q.E.D.: Theorem 24].Equation (160) provides one standard form for the representation of operator products in the sense of logarithmic representation. Consequently, BX-module is associated with the preinfinitesimal generator.


## 5.2. Formulation of Rotation Group

The application example ofBX-module is provided. The concept of BX-module is generally enough to provide a foundation of the conventional bounded formulation of Lie algebras (for a textbook, see [43]). In other words, by means of BX-module, the intersection of the Banach algebra (including only bounded operators) and the extracted bounded part of the Lie algebra (generally including unbounded operators) is shown. More precisely, using BX-module, the bounded part is extracted from unbounded angular momentum operators. The extracted bounded parts are utilized to formulate the rotation group with incorporating the unboundedness of angular momentum algebra.The mathematical foundation of rotation group is demonstrated [14]. Although the evolution parameter in this paper is denoted by t,s∈−T,+T, it is more likely to be denoted by θ,σ∈−Θ,+Θ, because the evolution parameter in the present case means the rotation angle. The rotation group is generated by the angular momentum operator (for textbooks, see Refs. [44, 45]). The angular momentum operator includes a differential operator, as represented by
(162)L−ihr×∇,where ℏ is a real constant called the Dirac constant. The appearance of differential operator ∇ in the representation of L is essential. The operator ∇ is an unbounded operator for example in a Hilbert space L2R3, while it must be treated as a bounded operator in terms of establishing an algebraic ring structure. Furthermore, the operator boundedness is also indispensable for some important formulae such as the Baker-Campbell-Hausdorff formula and the Zassenhaus formula to be valid. In general, the exponential of unbounded operators cannot be represented by the power series expansion (cf. the Yosida approximation in a typical proof of the Hille-Yosida theorem; e.g., see Ref. [22]).Letℝ3 be the three-dimensional spatial coordinate spanned by the standard orthogonal axes, x, y, and z. The angular momentum operator L is considered in L2ℝ3. The angular momentum operator
(163)L=Lx,Ly,Lz,consists of x, y, and z components
(164)Lx=ihy∂z−z∂y,Ly=ihz∂x−x∂z,Lz=ihy∂y−z∂x,respectively. The commutation relations
(165)Lx,Ly=ihLz,Ly,Lz,ihLx,Lz,Lx=ihLy,are true, where Li,Lj≔LiLj−LjLi denotes a commutator product i,j=x,y,z. The commutation of angular momentum operators arises from the commutation relations of the canonical quantization
(166)x,px=y,py=z,pz=ih,y,px=y,pz=z,py=x,py=x,pz=0.Indeed, the momentum operatorp=px,py,pz is represented by p=ih∂x,∂y,∂z in quantum mechanics. It is remarkable that the commutation is always true for the Newtonian mechanics; i.e., x,px=y,py=z,pz=0 is true in addition to y,px=y,pz=z,px=z,py=x,py=x,pz=0.Let a set of all bounded operators onL2ℝ3 be denoted by BL2ℝ3. A set of operators Lk;k=x,y,z or iLk/h;k=x,y,z with the commutation relation (165) is regarded as the Lie algebra. In particular α^Lx+β^Ly+γ^Lz;α^,β^,γ^∈ℂ forms a vector space over the complex number field, while αiLx/h+βiLy/h+γiLz/h;α,β,γ∈ℝ is a vector space over the real number field. It is possible to associate the real numbers α, β, and γ with the Euler angles (for example, see Ref. [46]). The second term of the right hand side of
(167)ri∂jrk∂l=ri∂jrk∂l+rirk∂j∂l,disappears as far as the commutator product Li,Lj is concerned, where ri is equal to i, and i,j,k,l=x,y, or z satisfy i≠j and k≠l. This fact is a key to justify the algebraic ring structure of Lk;k=x,y,z. On the other hand, although Lk is assumed to be bounded on L2ℝ3 in the typical treatment of the Lie algebra, it is not the case for the angular momentum algebra because of the appearance of differential operators in their definitions. From a geometric point of view, the range space RLk⊂L2ℝ3 strictly includes the domain space DLk; i.e., there is no guarantee for any u∈L2ℝ3 and a certain positive M∈ℝ to satisfy LkuL2ℝ3≤MuL2ℝ3. In order to establish iLk/h;k=x,y,z as the Lie algebra, it is necessary to show
(168)±iLk/h=±ri∂rj−rj∂ri,as an infinitesimal generator in Lr2ℝ3, where i,j,k=x,y,z satisfies i≠j≠k. As for the angular momentum operator, the t-independent assumption for operators K and L in Theorem 23 and Theorem 24 is satisfied. Note that t-independence assumes a kind of commutation relation.Theorem 25.
(unbounded formulation of rotation group). Letri be either x,y, or z. For i≠j, an operator ±ri∂rj with its domain space Hr2ℝ3 is an infinitesimal generator in Lr2ℝ3. Consequently, the angular momentum operators
(169)±iLx/h=±y∂z−z∂y,±iLy/h=±z∂x−x∂z,±iLz/h=±x∂y−y∂x,are infinitesimal generators in Lr2ℝ3.
Proof. The proof consists of three steps.
Step 1. ∂rk as an infinitesimal generator.
Lemma 26. For rk equal to x,y, or z, an operator ∂rk with its domain H1ℝ3 is an infinitesimal generator in Lr2ℝ3.
Proof of Lemma 26 The operator∂x is known as the infinitesimal generator of the first order hyperbolic type partial differential equations. For a complex number λ satisfying Reλ>0, let us consider a differential equation
(170)∂xux=λux−fx,in L2ℝ, and
(171)ux=−∫x∞expλx,ξfξdξ,satisfies the equation. According to the Schwarz inequality,
(172)∫−∞+∞ux2dx=∫−∞+∞∫x∞expλx,ξfξdξ2dx≤∫ξixi∫x∞expReλx−ξ2expReλx−ξ2fξdξ2dx≤∫ξixi∫x∞expReλx−ξdξ∫x∞expReλx−ξfξ2dξdx,is obtained, because eλ/22=eReλ/22eiImλ/22≤eReλ is valid if Reλ>0. Here, the equality
(173)∫x∞expReλx−ξdξ=∫0∞exp−Reλξdξ=1Reλ,is positive valued if Reλ>0. Its application leads to
(174)∫−∞+∞ux2dx≤1Reλ∫−∞+∞∫x∞expReλx−ξfξ2dξdx≤1Reλ∫−∞+∞∫−∞ξexpReλx−ξdxfξ2dξ.
Further application of the equality(175)∫−∞ξexpReλx−ξdx=∫−∞0expReλxdx=1Reλ,results in
(176)∫−∞+∞ux2dx≤1Reλ2∫−∞+∞fξ2dξ,and therefore
(177)λI−∂x−1fL2ℝ≤1Reλ2fL2ℝ.
That is, forReλ>0,
(178)λI−∂x−1≤1Reλ,is valid. The surjective property of λI−∂k is seen by the unique existence of solution ux∈L2ℝ for the initial value problem of equation (170).
A semigroup is generated by taking a subset of the complex plane as(179)Ω=λ∈ℂ,λ=λ¯,where Ω is included in the resolvent set of ∂x. For λ∈Ω,λI−∂x−1 exists, and
(180)λI−∂x−n≤1Reλn,is obtained. Consequently, according to the Lumer-Phillips theorem [38, 40] for the generation of quasi contraction semigroup, ∂x with the domain space H1ℝ3 is confirmed to be an infinitesimal generator in L2ℝ. The similar argument is valid to ∂y and ∂z. By considering x,y,z∈ℝ3, ∂k with k=x,y,z is the infinitesimal generators in L2ℝ3 [Q.E.D.: Lemma 26].
Step 2. irkI as an infinitesimal generator.
Lemma 27. Let rk be either x,y, or z. Let I be the identity operator of L2ℝ. An operator irkI is an infinitesimal generator in L2ℝ.
Proof of Lemma 27 For anyw∈ℂ, it is possible to define the exponential function by the convergent power series: ew=∑j=0∞wj/j!, so that
(181)eitrkI=∑j=0∞1j!itrkIj,is well-defined for t,rk∈ℝ. This fact is ensured by the boundedness of the identity operator I, although rkI and irkI are not bounded operators in L2ℝ if the standard L2-norm is equipped. It is sufficient for irkI to be the preinfinitesimal generator.
For an arbitraryrk∈ℝ, an operator irkI with its domain L2ℝ is the infinitesimal generator in L2ℝ; indeed, the spectral set is on the imaginary axis of the complex plane, and the unitary operator is generated as
(182)∫eitrkIu2drk=∫u2drk.
Consequently, the operatorirkI is treated as an infinitesimal generator in L2ℝ and therefore in L2ℝ3 [Q.E.D.: Lemma 27].
Step 3. ±ri∂rj−rj∂ri as an infinitesimal generator. Let i≠j be satisfied for i,j=x,y,z. Since r±t−s∂rj is well-defined (cf. Lemma 2) with the domain space Hr1ℝ3, its logarithmic representation is obtained by
(183)±iri∂rj=iriI−κe±s−t∂rj∂tLoge±t−s∂rj+κI,where κ≠0 is a certain complex number. According to Theorem 24, the product between iriI and ±∂rj is represented by
(184)±iri∂rj=iriI−κe±s−t∂rj∂tLoge±t−s∂rj+κI.
Using the commutation andt-independence of riI, it leads to the logarithmic representation
(185)±ri∂rj=I−κe±s−t∂rj∂triLoge±t−s∂rj+κI,without the loss of generality. The domain space of riI is equal to L2ℝ3, as eiriI is represented by the convergent power series in Lr2ℝ3. The half plane λ∈ℂ;Reλ>0 is included in the resolvent set of ±ri∂rj. Consequently, for t,ri∈ℝ, the existence of e±tri∂rj directly follows from the confirmed existence of e±t∂rj. Being equipped with the domain space Hr1ℝ3,±ri∂rj is the infinitesimal generator in Lr2ℝ3.
The preinfinitesimal generator property of sum is also understood by theBX-module property. The sum between ri∂ri and −rj∂ri is represented by
(186)I+κe+s−t∂rj∂triLoge+t−s∂rj+κI−I+κe−s−t∂ri∂trjLoge−t−s∂ri+κI=I+κe+s−t∂rj∂triLoge+s−t∂rj+κI−rjLoge−t−s∂ri+κI−κe+s−t∂rj−κe−s−t∂ri∂trjLoge−t−s∂ri+κI=I+κe+s−t∂rj∂triLoge+t−s∂rj+κI−Loge−t−s∂ri+κI−I+κe+s−t∂ri∂tri−rjLoge−t−s∂ri+κI−κe+s−t∂rj−κe−s−t∂ri∂trjLoge−t−s∂ri+κI,where all the three terms in the right hand side are of the form
(187)Lt∂triLogUt,s+Kt,whose preinfinitesimal generator properties are proved similarly to Lemma 26. In particular, the first term in the right hand side can be reduced to the above form with L=1 and t-dependent Kt, the parts corresponding to Lt are strongly continuous, and ri is independent of t. After having an integral of equation (186) in terms of t, each term is regraded as a bounded operator on Lr2ℝ3. Consequently, for i≠j, the application of Lemma 26 leads to the fact that
(188)±ri∂rj−rj∂ri,with its domain space Hr1ℝ3 is the infinitesimal generator in Lr2ℝ3 [Q.E.D.: Theorem 25].Corollary 28.
(collective renormalization). Fort,s∈−T,+T, let Vkt,s with k=x,y,z in Lr2ℝ3 be generated by iLk/h. For a certain complex constant ≠0, the angular momentum operator ±iLk/h with k=x,y,z is represented by the logarithm
(189)±iLk/h=±I+κVks,t∂tLogVkt,s+κI,and the corresponding evolution operator is expanded by the convergent power series
(190)Vkt,s=eLogVkt,s+κI−κI=∑n−0∞1n!LogVkt,s+κIn−κI=1−κI+∑n−0∞1n!LogVkt,s+κIn,where LogVkt,s+κI is bounded on Lr2ℝ3, although Lk is unbounded in Lr2ℝ3.
Proof. The group Vkt,s with k=x,y,z is generated by the infinitesimal generator iLk/h in Lr2ℝ3. This fact leads to the logarithmic representation
(191)iLk/h=I+κVks−t∂tLogVkt,s+κI,where κ≠0 is a certain complex constant. The relation Vkt,s+κI=eLogVkt,s+κI admits the power series expansion of Vkt,s [Q.E.D.: Corollary 28].Equation (189) shows a convergent power series representation for the rotation group. Let us call the representation shown in equation (189) the collective renormalization (cf. renormalized evolution equation in Corollary 9), in which a detailed degree of freedom ri∂rj is switched to a collective degree of freedom Lk. According to the collective renormalization, the evolution problem is studied by beginning with the bounded evolution operator Vks,t and the related bounded infinitesimal generator LogVkt,s+κI. In a more mathematical sense, the collective renormalization plays a role of simplifying the representation. Equation (190) ensures the validity of convergent power series expansions used in operator algebras even if they include unbounded operators.


## 6. Concluding Remarks

### 6.1. Template of Solvable Nonlinear Equations

The utility of the logarithmic representation is found in a formal discussion. The derivative of the logarithmic representation is formally represented by(192)∂tLogv=v′v−1,where v is a function of t, and the notation ′ denotes the differentiation along the t-direction. Since the logarithmic derivative ∂tLogv corresponds to the infinitesimal generator if v is the evolution operator, this equality shows the relation between the infinitesimal generator ∂tLogv and the evolution operator v. Let v be a known function (possibly a solution of linear equation) and u be an unknown function of t (uv−1 be a solution of another equation and of possibly a nonlinear equation). The Leibnitz rule reads
(193)uv−1′=v′v−1−u′u−1uv−1.Bothv′v−1 and u′u−1 are regarded as the logarithmic derivative for t-direction. The change of the evolution direction simply requires to fix u=∂xv, and then, uv−1 is regarded as the logarithmic derivative for x direction. As seen in the case of the Cole-Hopf transform, equation (193) being equivalent to the Burger’s equation in case of the Cole-Hopf transform provides one abstract template for nonlinear evolution equations, which can be analyzed as the linear problem. If the Cole-Hopf transform ψ=v′v−1 is combined with the Miura transform w=ψ′+ψ2, the higher order version of equation (192)
(194)w=v″v−1,is obtained [15]. In this way, the logarithmic representation provides templates of solvable nonlinear equations, which can be reduced to linear equations.
(i)
Related Topics. As for the applicability of the theory, the conditions to obtain the logarithmic representation (conditions shown in Section 2.2) are not so restrictive; indeed, they can be satisfied by C0-semigroups generated by xi-independent infinitesimal generators. The most restrictive condition to obtain the logarithmic representation is the commutation between Kxi and Uxi,ξi. Such a commutation is trivially satisfied by xi-independent Kxi=K and also satisfied when the variable xi is separable (i.e., for an integrable function gxi,Kxi=gxiK. In this sense, the operator specified in Theorem 16 corresponds to a moderate generalization of xi-independent infinitesimal generators. The summary is demonstrated along with the related topics.As for the applicability of the theory, the conditions to obtain the logarithmic representation (conditions shown in Section 2.2) are not so restrictive; indeed, they can be satisfied by C0-semigroups generated by xi-independent infinitesimal generators. The most restrictive condition to obtain the logarithmic representation is the commutation between Kxi and Uxi,ξi. Such a commutation is trivially satisfied by xi-independent Kxi=K and also satisfied when the variable xi is separable (i.e., for an integrable function gxi,Kxi=gxiK. In this sense, the operator specified in Theorem 16 corresponds to a moderate generalization of xi-independent infinitesimal generators. The summary is demonstrated along with the related topics.As for the applicability of the theory, the conditions to obtain the logarithmic representation (conditions shown in Section 2.2) are not so restrictive; indeed, they can be satisfied by C0-semigroups generated by xi-independent infinitesimal generators. The most restrictive condition to obtain the logarithmic representation is the commutation between Kxi and Uxi,ξi. Such a commutation is trivially satisfied by xi-independent Kxi=K and also satisfied when the variable xi is separable (i.e., for an integrable function gxi,Kxi=gxiK. In this sense, the operator specified in Theorem 16 corresponds to a moderate generalization of xi-independent infinitesimal generators. The summary is demonstrated along with the related topics.(ii)
Time Reversal Symmetry. Let the existence of negative time evolution be a kind of time reversal symmetry. Note that this kind of symmetry is true for linear wave equations but false for linear heat equations. The logarithmic representation of infinitesimal generators has been originally obtained for the invertible evolution operators, and it is generalized to noninvertible evolution operators. Under the validity of boundedness of Uxi,ξi on X, the removal of invertible criterion is essentially realized by the introduction of nonzero κ∈ℂ. On the other hand, the indispensable conditions for obtaining this kind of logarithmic representations are the boundedness of the spectral set of Uxi,ξi and the commutation assumption, where the bounded interval −L≤xi,ξi≤L is also necessary. Consequently, the time-reversal symmetry is recovered for the regularized evolution operator if xi is equal to x0. In the same way, a similar concept to spatial reversal symmetry being defined by the negative evolution can be recovered and violated by taking xi≠x0.(iii)
Regularity. The recovery of local time-reversal symmetry is associated with the regularity of the solution. The concept of regularized trajectory, whose regularity is similar to that of the analytic semigroups (for a textbook, see Ref. [22]) at the least, is true for regularized evolution operators.(iv)
Nonlinearity. For obtaining the logarithmic representation, the operator Uxi,ξi can be either linear or nonlinear semigroup. The nonlinearity of semigroup can appear simply by altering the evolution direction under a suitable identification between the infinitesimal generator and the evolution operator. In particular, the relation between evolution operator and its infinitesimal generator is essentially similar to the Cole-Hopf transform.(v)
The Self-Adjointness. The results obtained for a BX-module does not require the self-adjointness of the operator so that it opens up a way to have a full-complex analysis (neither real nor pure imaginary analysis) for a class of unbounded operators in association with the operator algebra. It is worth noting that the obtained algebraic structure corresponds to a generalization of “perturbation theory for semigroups of operators [42].”(vi)
Discrete Property. For example, in case of two-dimensional space-time distribution, let the C0-semigroup for x0 direction exists for a Cauchy problem:
(195)∂x0Ux0,ξ0u0=Kx0Ux0,ξ0u0,∂x0ax0,ξ0=∂x0LogUx0,ξ0+κI,in X0≔L2−L,L, where a certain complex number κ is taken from the resolvent set of Uxi,ξi. Furthermore, let the same equation possible to be written as
(196)∂x1Vx1ξ1v0=oKx1Vx1ξ1v0,∂x1αx1ξ1=o∂x1LogVx1ξ1+κI,in X1≔L2−T,T. In this situation, using αx1,ξ1 instead of αx0,ξ0, the corresponding dynamical system holds a discrete trajectory in X0 (for an illustration, see Figure 2 of Ref. [12]). Indeed, the trajectory is L2 function with respect to x0 and C0 function with respect to x1. That is, the relativistic treatment naturally leads to the discrete evolution (for a theory including the discrete evolution, see the variational method of abstract evolution equation [47, 48]). The discrete evolution to the t direction (x0 direction), which can be obtained by altering the evolution direction, is expected to be useful to analyze the stochastic differential equations within the semigroup theory of operators.


## 6.1. Template of Solvable Nonlinear Equations

The utility of the logarithmic representation is found in a formal discussion. The derivative of the logarithmic representation is formally represented by(192)∂tLogv=v′v−1,where v is a function of t, and the notation ′ denotes the differentiation along the t-direction. Since the logarithmic derivative ∂tLogv corresponds to the infinitesimal generator if v is the evolution operator, this equality shows the relation between the infinitesimal generator ∂tLogv and the evolution operator v. Let v be a known function (possibly a solution of linear equation) and u be an unknown function of t (uv−1 be a solution of another equation and of possibly a nonlinear equation). The Leibnitz rule reads
(193)uv−1′=v′v−1−u′u−1uv−1.Bothv′v−1 and u′u−1 are regarded as the logarithmic derivative for t-direction. The change of the evolution direction simply requires to fix u=∂xv, and then, uv−1 is regarded as the logarithmic derivative for x direction. As seen in the case of the Cole-Hopf transform, equation (193) being equivalent to the Burger’s equation in case of the Cole-Hopf transform provides one abstract template for nonlinear evolution equations, which can be analyzed as the linear problem. If the Cole-Hopf transform ψ=v′v−1 is combined with the Miura transform w=ψ′+ψ2, the higher order version of equation (192)
(194)w=v″v−1,is obtained [15]. In this way, the logarithmic representation provides templates of solvable nonlinear equations, which can be reduced to linear equations.
(i)
Related Topics. As for the applicability of the theory, the conditions to obtain the logarithmic representation (conditions shown in Section 2.2) are not so restrictive; indeed, they can be satisfied by C0-semigroups generated by xi-independent infinitesimal generators. The most restrictive condition to obtain the logarithmic representation is the commutation between Kxi and Uxi,ξi. Such a commutation is trivially satisfied by xi-independent Kxi=K and also satisfied when the variable xi is separable (i.e., for an integrable function gxi,Kxi=gxiK. In this sense, the operator specified in Theorem 16 corresponds to a moderate generalization of xi-independent infinitesimal generators. The summary is demonstrated along with the related topics.As for the applicability of the theory, the conditions to obtain the logarithmic representation (conditions shown in Section 2.2) are not so restrictive; indeed, they can be satisfied by C0-semigroups generated by xi-independent infinitesimal generators. The most restrictive condition to obtain the logarithmic representation is the commutation between Kxi and Uxi,ξi. Such a commutation is trivially satisfied by xi-independent Kxi=K and also satisfied when the variable xi is separable (i.e., for an integrable function gxi,Kxi=gxiK. In this sense, the operator specified in Theorem 16 corresponds to a moderate generalization of xi-independent infinitesimal generators. The summary is demonstrated along with the related topics.As for the applicability of the theory, the conditions to obtain the logarithmic representation (conditions shown in Section 2.2) are not so restrictive; indeed, they can be satisfied by C0-semigroups generated by xi-independent infinitesimal generators. The most restrictive condition to obtain the logarithmic representation is the commutation between Kxi and Uxi,ξi. Such a commutation is trivially satisfied by xi-independent Kxi=K and also satisfied when the variable xi is separable (i.e., for an integrable function gxi,Kxi=gxiK. In this sense, the operator specified in Theorem 16 corresponds to a moderate generalization of xi-independent infinitesimal generators. The summary is demonstrated along with the related topics.(ii)
Time Reversal Symmetry. Let the existence of negative time evolution be a kind of time reversal symmetry. Note that this kind of symmetry is true for linear wave equations but false for linear heat equations. The logarithmic representation of infinitesimal generators has been originally obtained for the invertible evolution operators, and it is generalized to noninvertible evolution operators. Under the validity of boundedness of Uxi,ξi on X, the removal of invertible criterion is essentially realized by the introduction of nonzero κ∈ℂ. On the other hand, the indispensable conditions for obtaining this kind of logarithmic representations are the boundedness of the spectral set of Uxi,ξi and the commutation assumption, where the bounded interval −L≤xi,ξi≤L is also necessary. Consequently, the time-reversal symmetry is recovered for the regularized evolution operator if xi is equal to x0. In the same way, a similar concept to spatial reversal symmetry being defined by the negative evolution can be recovered and violated by taking xi≠x0.(iii)
Regularity. The recovery of local time-reversal symmetry is associated with the regularity of the solution. The concept of regularized trajectory, whose regularity is similar to that of the analytic semigroups (for a textbook, see Ref. [22]) at the least, is true for regularized evolution operators.(iv)
Nonlinearity. For obtaining the logarithmic representation, the operator Uxi,ξi can be either linear or nonlinear semigroup. The nonlinearity of semigroup can appear simply by altering the evolution direction under a suitable identification between the infinitesimal generator and the evolution operator. In particular, the relation between evolution operator and its infinitesimal generator is essentially similar to the Cole-Hopf transform.(v)
The Self-Adjointness. The results obtained for a BX-module does not require the self-adjointness of the operator so that it opens up a way to have a full-complex analysis (neither real nor pure imaginary analysis) for a class of unbounded operators in association with the operator algebra. It is worth noting that the obtained algebraic structure corresponds to a generalization of “perturbation theory for semigroups of operators [42].”(vi)
Discrete Property. For example, in case of two-dimensional space-time distribution, let the C0-semigroup for x0 direction exists for a Cauchy problem:
(195)∂x0Ux0,ξ0u0=Kx0Ux0,ξ0u0,∂x0ax0,ξ0=∂x0LogUx0,ξ0+κI,in X0≔L2−L,L, where a certain complex number κ is taken from the resolvent set of Uxi,ξi. Furthermore, let the same equation possible to be written as
(196)∂x1Vx1ξ1v0=oKx1Vx1ξ1v0,∂x1αx1ξ1=o∂x1LogVx1ξ1+κI,in X1≔L2−T,T. In this situation, using αx1,ξ1 instead of αx0,ξ0, the corresponding dynamical system holds a discrete trajectory in X0 (for an illustration, see Figure 2 of Ref. [12]). Indeed, the trajectory is L2 function with respect to x0 and C0 function with respect to x1. That is, the relativistic treatment naturally leads to the discrete evolution (for a theory including the discrete evolution, see the variational method of abstract evolution equation [47, 48]). The discrete evolution to the t direction (x0 direction), which can be obtained by altering the evolution direction, is expected to be useful to analyze the stochastic differential equations within the semigroup theory of operators.



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