# On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis

Authors: Marat V. Markin
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

Given the abstract evolution equationy′(t)=Ay(t),  t∈R, withscalar type spectral operatorA in a complex Banach space, found are conditionsnecessary and sufficient for allweak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with anormal operatorA in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.


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## Body

## 1. Introduction

We find conditions on ascalar type spectral operator A in a complex Banach space necessary and sufficient for allweak solutions of the evolution equation(1)y′t=Ayt,t∈R,which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with anormal operatorA in a complex Hilbert space is obtained immediately as a particular case. We also prove the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.The found results develop those of paper [1], where similar consideration is given to the strong differentiability of the weak solutions of the equation(2)y′t=Ayt,t≥0,on [0,∞) and (0,∞).Definition 1 (weak solution).
LetA be a densely defined closed linear operator in a Banach space X and I be an interval of the real axis R. A strongly continuous vector function y:I→X is called aweak solution of the evolution equation(3)y′t=Ayt,t∈I,if, for any g∗∈D(A∗), (4)ddtyt,g∗=yt,A∗g∗,t∈I, where D(·) is thedomain of an operator, A∗ is the operatoradjoint to A, and ·,· is thepairing between the space X and its dual X∗ (cf. [2]).Remarks 2.
(i)
Due to theclosedness of A, a weak solution of (3) can be equivalently defined to be a strongly continuous vector function y:I↦X such that, for all t∈I, (5)∫t0tysds∈DAandyt=yt0+A∫t0tysds, wheret0 is an arbitrary fixed point of the interval I, and is also called amild solution (cf. [3, Ch. II, Definition 6.3], see also [4, Preliminaries]). (ii)
Such a notion ofweak solution, which need not be differentiable in the strong sense, generalizes that ofclassical one, strongly differentiable on I and satisfying the equation in the traditional plug-in sense, the classical solutions being precisely the weak ones strongly differentiable on I. (iii)
As is easily seeny:R→X is a weak solution of (1)iff(6)y+t≔yt,t≥0, is a weak solution of (2) and(7)y-t≔y-t,t≥0, is a weak solution of the equation(8)y′t=-Ayt,t≥0. (iv)
When a closed densely defined linear operatorA in a complex Banach space X generates a strongly continuous group Ttt∈R of bounded linear operators (see, e.g., [3, 5]), i.e., the associatedabstract Cauchy problem (ACP)(9)y′t=Ayt,t∈R,y0=f iswell-posed (cf. [3, Ch. II, Definition 6.8]), the weak solutions of (1) are the orbits(10)yt=Ttf,t∈R, withf∈X (cf. [3, Ch. II, Proposition 6.4], see also [2, Theorem]), whereas the classical ones are those with f∈D(A) (see, e.g., [3, Ch. II, Proposition 6.3]). (v)
In our discourse, the associatedACP may beill-posed, i.e., the scalar type spectral operator A need not generate a strongly continuous group of bounded linear operators (cf. [6]).


## 2. Preliminaries

Here, for the reader’s convenience, we outline certain essential preliminaries.Henceforth, unless specified otherwise,A is supposed to be ascalar type spectral operator in a complex Banach space (X,·) with strongly σ-additivespectral measure (theresolution of the identity) EA(·) assigning to each Borel set δ of the complex plane C a projection operator EA(δ) on X and having the operator’sspectrumσ(A) as itssupport [7, 8].Observe that, in a complex finite-dimensional space, the scalar type spectral operators are all linear operators on the space, for which there is aneigenbasis (see, e.g., [7, 8]) and, in a complex Hilbert space, the scalar type spectral operators are precisely all those that are similar to thenormal ones [9].Associated with a scalar type spectral operator in a complex Banach space is theBorel operational calculus analogous to that for anormal operator in a complex Hilbert space [7, 8, 10, 11], which assigns to any Borel measurable function F:σ(A)→C a scalar type spectral operator (11)FA≔∫σAFλdEAλ(see [7, 8]).In particular,(12)An=∫σAλndEAλ,n∈Z+,(Z+≔0,1,2,… is the set ofnonnegative integers, A0≔I, I is theidentity operator on X), and(13)ezA≔∫σAezλdEAλ,z∈C.The properties of thespectral measure andoperational calculus, exhaustively delineated in [7, 8], underlie the entire subsequent discourse. Here, we underline a few facts of particular importance.Due to itsstrong countable additivity, the spectral measure EA(·) isbounded [8, 12], i.e., there is such an M>0 that, for any Borel set δ⊆C,(14)EAδ≤M.Observe that the notation · is used here to designate the norm in the space L(X) of all bounded linear operators on X. We adhere to this rather conventional economy of symbols in what follows also adopting the same notation for the norm in the dual space X∗.For anyf∈X and g∗∈X∗, thetotal variation measure  v(f,g∗,·) of the complex-valued Borel measure EA·f,g∗ is afinite positive Borel measure with(15)vf,g∗,C=vf,g∗,σA≤4Mfg∗(see, e.g., [13, 14]).Also (Ibid.), for a Borel measurable functionF:C→C, f∈D(F(A)), g∗∈X∗, and a Borel set δ⊆C,(16)∫δFλdvf,g∗,λ≤4MEAδFAfg∗.In particular, for δ=σ(A), EA(σ(A))=I and(17)∫σAFλdvf,g∗,λ≤4MFAfg∗.Observe that the constant M>0 in (15)–(17) is from (14).Further, for a Borel measurable functionF:C→[0,∞), a Borel set δ⊆C, a sequence Δnn=1∞ of pairwise disjoint Borel sets in C, and f∈X, g∗∈X∗,(18)∫δFλdvEA⋃n=1∞Δnf,g∗,λ=∑n=1∞∫δ∩ΔnFλdvEAΔnf,g∗,λ.Indeed, since, for any Borel sets δ,σ⊆C, (19)EAδEAσ=EAδ∩σ[7, 8], for the total variation measure, (20)vEAδf,g∗,σ=vf,g∗,δ∩σ.Whence, due to thenonnegativity of F(·) (see, e.g., [15]),(21)∫δFλdvEA⋃n=1∞Δnf,g∗,λ=∫δ∩∪n=1∞ΔnFλdvf,g∗,λ=∑n=1∞∫δ∩ΔnFλdvf,g∗,λ=∑n=1∞∫δ∩ΔnFλdvEAΔnf,g∗,λ.The following statement, allowing characterizing the domains of Borel measurable functions of a scalar type spectral operator in terms of positive Borel measures, is fundamental for our discourse.Proposition 3 ([16, Proposition 3.1]).
LetA be a scalar type spectral operator in a complex Banach space (X,·) with spectral measure EA(·) and F:σ(A)→C be a Borel measurable function. Then f∈D(F(A)) iff (i)
for eachg∗∈X∗, ∫σAFλdvf,g∗,λ<∞;(ii)
supg∗∈X∗∣∥g∗∥=1⁡∫λ∈σA∣Fλ>nFλdvf,g∗,λ→0,  n→∞,
wherev(f,g∗,·) is the total variation measure of EA·f,g∗.The succeeding key theorem provides a description of the weak solutions of (2) with a scalar type spectral operator A in a complex Banach space.Theorem 4 ([16, Theorem 4.2] with T=∞).
LetA be a scalar type spectral operator in a complex Banach space (X,·). A vector function y:[0,∞)→X is a weak solution of (2) iff there is an f∈⋂t≥0DetA such that (22)yt=etAf,t≥0, the operator exponentials understood in the sense of the Borel operational calculus (see (13)).Remark 5.
Theorem4 generalizes [17, Theorem 3.1], its counterpart for a normal operator A in a complex Hilbert space.We also need the following characterizations of a particular weak solution’s of (2) with a scalar type spectral operator A in a complex Banach space being strongly differentiable on a subinterval I of [0,∞).Proposition 6 ([1, Proposition 3.1] with T=∞).
Letn∈N and I be a subinterval of [0,∞). A weak solution y(·) of (2) is n times strongly differentiable on I iff (23)yt∈DAn,t∈I, in which case (24)ykt=Akyt,k=1,…,n,t∈I.Subsequently, the frequent terms“spectral measure” and“operational calculus” are abbreviated tos.m. ando.c., respectively.


## 3. General Weak Solution

Theorem 7 (general weak solution).
LetA be a scalar type spectral operator in a complex Banach space (X,·). A vector function y:R→X is a weak solution of (1) iff there is an f∈⋂t∈RD(etA) such that(25)yt=etAf,t∈R,the operator exponentials understood in the sense of the Borel operational calculus (see (13)).Proof.
As is noted in the Introduction,y:R→X is a weak solution of (1)iff(26)y+t≔yt,t≥0,is a weak solution of (2) and(27)y-t≔y-t,t≥0,is a weak solution of (8).
Applying Theorem4, to y+(·) and y-(·), we infer that this is equivalent to the fact(28)yt=etAf,t∈R,with  somef∈⋂t∈RDetA.Remarks 8.
(i)
More generally, Theorem4 and its proof can be easily modified to describe in the same manner all weak solution of (3) for an arbitrary interval I of the real axis R. (ii)
Theorem7 implies, in particular,
(a)
that the subspace⋂t∈RD(etA) of all possible initial values of the weak solutions of (1) is the largest permissible for the exponential form given by (25), which highlights the naturalness of the notion of weak solution; (b)
that associatedACP (9), whenever solvable, is solvableuniquely. (iii)
Observe that the initial-value subspace⋂t∈RD(etA) of (1), containing the dense in X subspace ⋃α>0EA(Δα)X, where (29)Δα≔λ∈C∣λ≤α,α>0, which coincides with the classE0(A) ofentire vectors of A ofexponential type [18], isdense in X as well. (iv)
When a scalar type spectral operatorA in a complex Banach space generates a strongly continuous group T(t)t∈R of bounded linear operators,(30)Tt=etAandDetA=X,t∈R, [6], and hence, Theorem 7 is consistent with the well-known description of the weak solutions for this setup (see (10)). (v)
Clearly, the initial-value subspace⋂t∈RDetA of (1) is narrower than the initial-value subspace ⋂t≥0D(etA) of (2) and the initial-value subspace ⋂t≥0D(et(-A))=⋂t≤0D(etA) of (8); in fact it is the intersection of the latter two.


## 4. Differentiability of a Particular Weak Solution

Here, we characterize a particular weak solution’s of (1) with a scalar type spectral operator A in a complex Banach space being strongly differentiable on a subinterval I of R.Proposition 9 (differentiability of a particular weak solution).
Letn∈N and I be a subinterval of R. A weak solution y(·) of (1) is n times strongly differentiable on I iff(31)yt∈DAn,t∈I,in which case, (32)ykt=Akyt,k=1,…,n,t∈I.Proof.
The statement immediately follows from the prior theorem and Proposition6 applied to (33)y+t≔ytandy-t≔y-t,t≥0,for an arbitrary weak solution y(·) of (1).Remark 10.
Observe that, as well as for Proposition6, for n=1, the subinterval I can degenerate into a singleton.Inductively, we immediately obtain the following analog of [1, Corollary 3.2].Corollary 11 (infinite differentiability of a particular weak solution).
LetA be a scalar type spectral operator in a complex Banach space (X,·) and I be a subinterval of R. A weak solution y(·) of (1) is strongly infinite differentiable on I (y(·)∈C∞(I,X)) iff, for each t∈I, (34)yt∈C∞A≔⋂n=1∞DAn, in which case (35)ynt=Anyt,n∈N,t∈I.


## 5. Infinite Differentiability of Weak Solutions

In this section, we characterize the strong infinite differentiability onR of all weak solutions of (1) with a scalar type spectral operator A in a complex Banach space.Theorem 12 (infinite differentiability of weak solutions).
LetA be a scalar type spectral operator in a complex Banach space (X,·) with spectral measure EA(·). Every weak solution of (1) is strongly infinite differentiable on R iff there exist b+>0 and b->0 such that the set σ(A)\Lb-,b+, where (36)Lb-,b+≔λ∈C∣Re⁡λ≤min0,-b-ln⁡ImλorRe⁡λ≥max⁡0,b+ln⁡Imλ, is bounded (see Figure 1).Figure 1Proof.
“If” part: suppose that there exist b+>0 and b->0 such that the set σ(A)\Lb-,b+ isbounded and let y(·) be an arbitrary weak solution of (1).
By Theorem7, (37)yt=etAf,t∈R,withsomef∈⋂t∈RDetA.Our purpose is to show that y(·)∈C∞R,X, which, by Corollary 11, is attained by showing that, for each t∈R,(38)yt∈C∞A≔⋂n=1∞DAn.Let us proceed by proving that, for any t∈R and m∈N(39)yt∈DAm via Proposition 3.
For anyt∈R, m∈N, and an arbitrary g∗∈X∗,(40)∫σAλmetReλdvf,g∗,λ=∫σA\Lb-,b+λmetReλdvf,g∗,λ+∫λ∈σA∩Lb-,b+∣-1<Reλ<1λmetReλdvf,g∗,λ+∫λ∈σA∩Lb-,b+∣Reλ≥1λmetReλdvf,g∗,λ+∫λ∈σA∩Lb-,b+∣Reλ≤-1λmetReλdvf,g∗,λ<∞.Indeed,(41)∫σA\Lb-,b+λmetReλdvf,g∗,λ<∞ and(42)∫λ∈σA∩Lb-,b+∣-1<Reλ<1λmetReλdvf,g∗,λ<∞ due to the boundedness of the sets(43)σA\Lb-,b+andλ∈σA∩Lb-,b+∣-1<Reλ<1, the continuity of the integrated function on C, and the finiteness of the measure v(f,g∗,·).
Further, for anyt∈R, m∈N, and an arbitrary g∗∈X∗,(44)∫λ∈σA∩Lb-,b+∣Reλ≥1λmetReλdvf,g∗,λ≤∫λ∈σA∩Lb-,b+∣Reλ≥1Reλ+ImλmetReλdvf,g∗,λsince,forλ∈σA∩Lb-,b+withReλ≥1,eb+-1Reλ≥Imλ;≤∫λ∈σA∩Lb-,b+∣Reλ≥1Reλ+eb+-1ReλmetReλdvf,g∗,λsince,inviewofReλ≥1,b+eb+-1Reλ≥Reλ;≤∫λ∈σA∩Lb-,b+∣Reλ≥1b+eb+-1Reλ+eb+-1ReλmetReλdvf,g∗,λ=b++1m∫λ∈σA∩Lb-,b+∣Reλ≥1emb+-1+tReλdvf,g∗,λsincef∈⋂t∈RDetA,byProposition3;<∞.Finally, for any t∈R, m∈N and an arbitrary g∗∈X∗,(45)∫λ∈σA∩Lb-,b+∣Reλ≤-1λmetReλdvf,g∗,λ≤∫λ∈σA∩Lb-,b+∣Reλ≤-1Reλ+ImλmetReλdvf,g∗,λsince,forλ∈σA∩Lb-,b+withReλ≤-1,eb--1-Reλ≥Imλ;≤∫λ∈σA∩Lb-,b+∣Reλ≤-1-Reλ+eb--1-ReλmetReλdvf,g∗,λsince,inviewof-Reλ≥1,b-eb--1-Reλ≥-Reλ;≤∫λ∈σA∩Lb-,b+∣Reλ≤-1b-eb--1-Reλ+eb--1-ReλmetReλdvf,g∗,λ=b-+1m∫λ∈σA∩Lb-,b+∣Reλ≤-1et-mb--1Reλdvf,g∗,λsincef∈⋂t∈RDetA,byProposition3;<∞.Also, for any t∈R, m∈N, and an arbitrary n∈N,(46)supg∗∈X∗∣g∗=1⁡∫λ∈σA∣λmetReλ>nλmetReλdvf,g∗,λ≤supg∗∈X∗∣g∗=1⁡∫λ∈σA\Lb-,b+∣λmetReλ>nλmetReλdvf,g∗,λ+supg∗∈X∗∣g∗=1⁡∫λ∈σA∩Lb-,b+∣-1<Reλ<1,λmetReλ>nλmetReλdvf,g∗,λ+supg∗∈X∗∣g∗=1⁡∫λ∈σA∩Lb-,b+∣Reλ≥1,λmetReλ>nλmetReλdvf,g∗,λ+supg∗∈X∗∣g∗=1⁡∫λ∈σA∩Lb-,b+∣Reλ≤-1,λmetReλ>nλmetReλdvf,g∗,λ→0,n→∞.Indeed, since, due to the boundedness of the sets(47)σA\Lb-,b+andλ∈σA∩Lb-,b+∣-1<Reλ<1and the continuity of the integrated function on C, the sets(48)λ∈σA\Lb-,b+∣λmetReλ>nand(49)λ∈σA∩Lb-,b+∣-1<Reλ<1,λmetReλ>nareempty for all sufficiently large n∈N, we immediately infer that, for any t∈R and m∈N,(50)limn→∞⁡supg∗∈X∗∣g∗=1⁡∫λ∈σA\Lb-,b+∣λmetReλ>nλmetReλdvf,g∗,λ=0and(51)limn→∞⁡supg∗∈X∗∣g∗=1⁡∫λ∈σA∩Lb-,b+∣-1<Reλ<1,λmetReλ>nλmetReλdvf,g∗,λ=0.Further, for any t∈R, m∈N, and an arbitrary n∈N,(52)supg∗∈X∗∣g∗=1⁡∫λ∈σA∩Lb-,b+∣Reλ≥1,λmetReλ>nλmetReλdvf,g∗,λasin44;≤supg∗∈X∗∣g∗=1⁡b++1m∫λ∈σA∩Lb-,b+∣Reλ≥1,λmetReλ>nemb+-1+tReλdvf,g∗,λsincef∈⋂t∈RDetA,by16;≤supg∗∈X∗∣g∗=1⁡b++1m4MEAλ∈σA∩Lb-,b+∣Reλ≥1,λmetReλ>nemb+-1+tAfg∗≤b++1m4MEAλ∈σA∩Lb-,b+∣Reλ≥1,λmetReλ>nemb+-1+tAfbythestrongcontinuityofthes.m.;→b++1m4MEA∅emb+-1+tAf=0,n→∞.Finally, for any t∈R, m∈N, and an arbitrary n∈N,(53)supg∗∈X∗∣g∗=1⁡∫λ∈σA∩Lb-,b+∣Reλ≤-1,λmetReλ>nλmetReλdvf,g∗,λasin45;≤supg∗∈X∗∣g∗=1⁡b-+1m∫λ∈σA∩Lb-,b+∣Reλ≤-1,λmetReλ>net-mb--1Reλdvf,g∗,λsincef∈⋂t∈RDetA,by16;≤supg∗∈X∗∣g∗=1⁡b-+1m4MEAλ∈σA∩Lb-,b+∣Reλ≤-1,λmetReλ>net-mb--1Afg∗≤b-+1m4MEAλ∈σA∩Lb-,b+∣Reλ≤-1,λmetReλ>net-mb--1Afbythestrongcontinuityofthes.m.;→b++1m4MEA∅et-mb--1Af=0,n→∞.By Proposition 3 and the properties of theo.c. (see [8, Theorem XVIII.2.11 (f)]), (40) and (46) jointly imply that, for any t∈R and m∈N,(54)f∈DAmetA,which further implies that, for each t∈R, (55)yt=etAf∈⋂n=1∞DAn≕C∞A.Whence, by Corollary 11, we infer that (56)y·∈C∞R,X,which completes the proof of the“if” part.
“Only if” part: let us prove this partby contrapositive assuming that, for any b+>0 and b->0, the set σ(A)\Lb-,b+ isunbounded. In particular, this means that, for any n∈N, unbounded is the set (57)σA\L2n-1,2n-1=λ∈σA∣-2n-1ln⁡Im⁡λ<Reλ<2n-1ln⁡Im⁡λ.Hence, we can choose a sequence of points λnn=1∞ in the complex plane as follows: (58)λn∈σA,n∈N,-2n-1ln⁡Im⁡λn<Reλn<2n-1ln⁡Im⁡λn,n∈N,λ0≔0,λn>max⁡n4,λn-1,n∈N.The latter implies, in particular, that the points λn, n∈N, aredistinct (λi≠λj, i≠j).
Since, for eachn∈N, the set (59)λ∈C∣-2n-1ln⁡Imλ<Reλ<2n-1ln⁡Imλ,λ>max⁡n4,λn-1isopen in C; along with the point λn, it contains anopen disk(60)Δn≔λ∈C∣λ-λn<εncentered at λn of some radius εn>0, i.e., for each λ∈Δn,(61)-2n-1ln⁡Imλ<Reλ<2n-1ln⁡Imλandλ>max⁡n4,λn-1.Furthermore, we can regard the radii of the disks to be small enough so that(62)0<εn<1n,n∈N,andΔi∩Δj=∅,i≠ji.e.,thedisksarepairwise  disjoint.Whence, by the properties of thes.m., (63)EAΔiEAΔj=0,i≠j,where 0 stands for thezero operator on X.
Observe also, that the subspacesEA(Δn)X, n∈N, arenontrivial since(64)Δn∩σA≠∅,n∈N,with Δn being anopen set in C.
By choosing a unit vectoren∈EA(Δn)X for each n∈N, we obtain a sequence enn=1∞ in X such that(65)en=1,n∈N,andEAΔiej=δijej,i,j∈N,where δij is theKronecker delta.
As is easily seen, (65) implies that the vectors en, n∈N, arelinearly independent.
Furthermore, there is anε>0 such that(66)dn≔dist⁡en,span⁡ei∣i∈N,i≠n≥ε,n∈N.Indeed, the opposite implies the existence of a subsequence dnkk=1∞ such that (67)dnk→0,k→∞.Then, by selecting a vector(68)fnk∈span⁡ei∣i∈N,i≠nk,k∈N,such that(69)enk-fnk<dnk+1k,k∈N,we arrive at(70)1=enksince,by65,EAΔnkfnk=0;=EAΔnkenk-fnk≤EAΔnkenk-fnkby14;≤Menk-fnk≤Mdnk+1k→0,k→∞,which is acontradiction proving (66).
As follows from theHahn-Banach Theorem, for any n∈N, there is an en∗∈X∗ such that(71)en∗=1,n∈N,andei,ej∗=δijdi,i,j∈N.Let us consider separately the two possibilities concerning the sequence of the real parts Reλnn=1∞: its beingbounded orunbounded.
First, suppose that the sequenceReλnn=1∞ isbounded, i.e., there is such an ω>0 that(72)Reλn≤ω,n∈N,and consider the element (73)f≔∑k=1∞k-2ek∈X,which is well defined since k-2k=1∞∈l1 (l1 is the space of absolutely summable sequences) and ek=1, k∈N (see (65)).
In view of (65), by the properties of thes.m.,(74)EA⋃k=1∞Δkf=fandEAΔkf=k-2ek,k∈N.For any t≥0 and an arbitrary g∗∈X∗,(75)∫σAetReλdvf,g∗,λby74;=∫σAetReλdvEA⋃k=1∞Δkf,g∗,λby18;=∑k=1∞∫σA∩ΔketReλdvEAΔkf,g∗,λby74;=∑k=1∞k-2∫σA∩ΔketReλdvek,g∗,λsince,forλ∈Δk,by72and62,Reλ=Reλk+Reλ-Reλk≤Reλk+λ-λk≤ω+εk≤ω+1;≤etω+1∑k=1∞k-2∫σA∩Δk1dvek,g∗,λ=etω+1∑k=1∞k-2vek,g∗,Δkby15;≤etω+1∑k=1∞k-24Mekg∗=4Metω+1g∗∑k=1∞k-2<∞.Also, for any t<0 and an arbitrary g∗∈X∗,(76)∫σAetReλdvf,g∗,λby74;=∫σAetReλdvEA⋃k=1∞Δkf,g∗,λby18;=∑k=1∞∫σA∩ΔketReλdvEAΔkf,g∗,λby74;=∑k=1∞k-2∫σA∩ΔketReλdvek,g∗,λsince,forλ∈Δk,by72and62,Reλ=Reλk-Reλk-Reλ≥Reλk-Reλk-Reλ≥-ω-εk≥-ω-1;≤e-tω+1∑k=1∞k-2∫σA∩Δk1dvek,g∗,λ=e-tω+1∑k=1∞k-2vek,g∗,Δkby15;≤e-tω+1∑k=1∞k-24Mekg∗=4Me-tω+1g∗∑k=1∞k-2<∞.Similarly to (75), for any t≥0 and an arbitrary n∈N,(77)supg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>netReλdvf,g∗,λ≤supg∗∈X∗∣g∗=1⁡etω+1∑k=1∞k-2∫λ∈σA∣etReλ>n∩Δk1dvek,g∗,λby74;=etω+1supg∗∈X∗∣g∗=1⁡∑k=1∞∫λ∈σA∣etReλ>n∩Δk1dvEAΔkf,g∗,λby18;=etω+1supg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>n1dvEA⋃k=1∞Δkf,g∗,λby74;=etω+1supg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>n1dvf,g∗,λby16;≤etω+1supg∗∈X∗∣g∗=14MEAλ∈σA∣etReλ>nfg∗≤4Metω+1EAλ∈σA∣etReλ>nfbythestrongcontinuityofthes.m.;→4Metω+1EA∅f=0,n→∞.Similarly to (76), for any t<0 and an arbitrary n∈N,(78)supg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>netReλdvf,g∗,λ≤supg∗∈X∗∣g∗=1⁡e-tω+1∑k=1∞k-2∫λ∈σA∣etReλ>n∩Δk1dvek,g∗,λby74;=e-tω+1supg∗∈X∗∣g∗=1⁡∑k=1∞∫λ∈σA∣etReλ>n∩Δk1dvEAΔkf,g∗,λby18;=e-tω+1supg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>n1dvEA⋃k=1∞Δkf,g∗,λby74;=e-tω+1supg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>n1dvf,g∗,λby16;≤e-tω+1supg∗∈X∗∣g∗=1⁡4MEAλ∈σA∣etReλ>nfg∗≤4Me-tω+1EAλ∈σA∣etReλ>nfbythestrongcontinuityofthes.m.;→4Me-tω+1EA∅f=0,n→∞.By Proposition 3, (75), (76), (77), and (78) jointly imply that(79)f∈⋂t∈RDetA, and hence, by Theorem 7,(80)yt≔etAf,t∈R, is a weak solution of (1).
Let(81)h∗≔∑k=1∞k-2ek∗∈X∗,the functional being well defined since {k-2}k=1∞∈l1 and ek∗=1, k∈N (see (71)).
In view of (71) and (66), we have(82)en,h∗=ek,k-2ek∗=dkk-2≥εk-2,k∈N.Hence,(83)∫σAλdvf,h∗,λby18asin75;=∑k=1∞k-2∫σA∩Δkλdvek,h∗,λsince,forλ∈Δk,by61,λ≥k4;≥∑k=1∞k-2k4vek,h∗,Δk≥∑k=1∞k2EAΔkek,h∗by65and82;≥∑k=1∞k2εk-2=∞.By Proposition 3, (83) implies that(84)y0=f∉DA, which, by Proposition 9 (n=1, I={0}) further implies that the weak solution y(t)=etAf, t∈R, of (1) is not strongly differentiable at 0.
Now, suppose that the sequenceReλnn=1∞ isunbounded.
Therefore, there is a subsequenceReλnkk=1∞ such that(85)Reλnk→∞orReλnk→-∞,k→∞. Let us consider separately each of the two cases.
First, suppose that(86)Reλnk→∞,k→∞ Then, without loss of generality, we can regard that(87)Reλnk≥k,k∈N.Consider the elements (88)f≔∑k=1∞e-nkReλnkenk∈Xandh≔∑k=1∞e-nk/2Reλnkenk∈X,well defined since, by (87),(89)e-nkReλnkk=1∞,e-nk/2Reλnkk=1∞∈l1and enk=1, k∈N (see (65)).
By (65),(90)EA⋃k=1∞Δnkf=fandEAΔnkf=e-nkReλnkenk,k∈N,and(91)EA⋃k=1∞Δnkh=handEAΔnkh=e-nk/2Reλnkenk,k∈N.For any t≥0 and an arbitrary g∗∈X∗,(92)∫σAetReλdvf,g∗,λby18asin75;=∑k=1∞e-nkReλnk∫σA∩ΔnketReλdvenk,g∗,λsince,forλ∈Δnk,by62,Reλ=Reλnk+Reλ-Reλnk≤Reλnk+λ-λnk≤Reλnk+1;≤∑k=1∞e-nkReλnketReλnk+1∫σA∩Δnk1dvenk,g∗,λ=et∑k=1∞e-nk-tReλnkvenk,g∗,Δnkby15;≤et∑k=1∞e-nk-tReλnk4Menkg∗=4Metg∗∑k=1∞e-nk-tReλnk<∞.Indeed, for all k∈N sufficiently large so that(93)nk≥t+1,in view of (87),(94)e-nk-tReλnk≤e-k.For any t<0 and an arbitrary g∗∈X∗,(95)∫σAetReλdvf,g∗,λby18asin75;=∑k=1∞e-nkReλnk∫σA∩ΔnketReλdvenk,g∗,λsince,forλ∈Δnk,by62,Reλ=Reλnk-Reλnk-Reλ≥Reλnk-Reλnk-Reλ≥Reλnk-1;≤∑k=1∞e-nkReλnketReλnk-1∫σA∩Δnk1dvenk,g∗,λ=e-t∑k=1∞e-nk-tReλnkvenk,g∗,Δnkby15;≤e-t∑k=1∞e-nk-tReλnk4Menkg∗=4Me-tg∗∑k=1∞e-nk-tReλnk<∞.Indeed, for all k∈N, in view of t<0,(96)nk-t≥nk≥1, and hence, in view of (87),(97)e-nk-tReλnk≤e-k.Similarly to (92), for any t≥0 and an arbitrary n∈N,(98)supg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>netReλdvf,g∗,λ≤supg∗∈X∗∣g∗=1⁡et∑k=1∞e-nk-tReλnk∫λ∈σA∣etReλ>n∩Δnk1dvenk,g∗,λ=etsupg∗∈X∗∣g∗=1⁡∑k=1∞e-nk/2-tReλnke-nk/2Reλk∫λ∈σA∣etReλ>n∩Δnk1dvenk,g∗,λsince,by87,thereisanL>0suchthate-nk/2-tReλnk≤L,k∈N;≤Letsupg∗∈X∗∣g∗=1⁡∑k=1∞e-nk/2Reλnk∫λ∈σA∣etReλ>n∩Δnk1dvenk,g∗,λby91;=Letsupg∗∈X∗∣g∗=1⁡∑k=1∞∫λ∈σA∣etReλ>n∩Δnk1dvEAΔnkh,g∗,λby18;=Letsupg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>n1dvEA⋃k=1∞Δnkh,g∗,λby91;=Letsupg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>n1dvh,g∗,λby16;≤Letsupg∗∈X∗∣g∗=1⁡4MEAλ∈σA∣etReλ>nhg∗≤4LMetEAλ∈σA∣etReλ>nhbythestrongcontinuityofthes.m.;→4LMetEA∅h=0,n→∞.Similar to (95), for any t<0 and an arbitrary n∈N,(99)supg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>netReλdvf,g∗,λ≤supg∗∈X∗∣g∗=1⁡e-t∑k=1∞e-nk-tReλnk∫λ∈σA∣etReλ>n∩Δnk1dvenk,g∗,λ=e-tsupg∗∈X∗∣g∗=1⁡∑k=1∞e-nk/2-tReλnke-nk/2Reλk∫λ∈σA∣etReλ>n∩Δnk1dvenk,g∗,λsince,by87,thereisanL>0suchthate-nk/2-tReλnk≤L,k∈N;≤Le-tsupg∗∈X∗∣g∗=1⁡∑k=1∞e-nk/2Reλnk∫λ∈σA∣etReλ>n∩Δnk1dvenk,g∗,λby91;=Le-tsupg∗∈X∗∣g∗=1⁡∑k=1∞∫λ∈σA∣etReλ>n∩Δnk1dvEAΔnkh,g∗,λby18;=Le-tsupg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>n1dvEA⋃k=1∞Δnkh,g∗,λby91;=Le-tsupg∗∈X∗∣g∗=1⁡∫λ∈σA∣etReλ>n1dvh,g∗,λby16;≤Le-tsupg∗∈X∗∣g∗=1⁡4MEAλ∈σA∣etReλ>nhg∗≤4LMe-tEAλ∈σA∣etReλ>nhbythestrongcontinuityofthes.m.;→4LMe-tEA∅h=0,n→∞.By Proposition 3, (92), (95), (98), and (99) jointly imply that(100)f∈⋂t∈RDetA, and hence, by Theorem 7,(101)yt≔etAf,t∈R, is a weak solution of (1).
Since, for anyλ∈Δn(k), k∈N, by (62), (87),(102)Reλ=Reλnk-Reλnk-Reλ≥Reλnk-Reλnk-Reλ≥Reλnk-εnk≥Reλnk-1nk≥k-1≥0 and, by (61),(103)Reλ<2nk-1ln⁡Imλ, we infer that, for any λ∈Δn(k), k∈N, (104)λ≥Imλ≥e2nkReλ≥e2nkReλnk-1/nk.Using this estimate, for the functional h∗∈X∗ defined by (81), we have(105)∫σAλdvf,h∗,λby18asin75;=∑k=1∞e-nkReλnk∫Δnkλdvenk,h∗,λ≥∑k=1∞e-nkReλnke2nkReλnk-1/nkvenk,h∗,Δnk=∑k=1∞e-2enkReλnkEAΔnkenk,h∗by87,65,and82;≥∑k=1∞e-2εenknk2=∞.By Proposition 3, (83) implies that(106)y0=f∉DA, which, by Proposition 9 (n=1, I={0}), further implies that the weak solution y(t)=etAf, t∈R, of (1) is not strongly differentiable at 0.
The remaining case of(107)Reλnk→-∞,k→∞, is symmetric to the case of(108)Reλnk→∞,k→∞, and is considered in absolutely the same manner, which furnishes a weak solution y(·) of (1) such that (109)y0∉DA, and hence, by Proposition 9 (n=1, I={0}), not strongly differentiable at 0.
With every possibility concerningReλnn=1∞ considered, we infer that assuming the opposite to the“if” part’s premise allows to find a weak solution of (1) on [0,∞) that is not strongly differentiable at 0, much less strongly infinite differentiable on R.
Thus, the proof by contrapositive of the“only if” part is complete and so is the proof of the entire statement.From Theorem12 and [1, Theorem 4.2], the latter characterizing the strong infinite differentiability of all weak solution of (2) on (0,∞), we also obtain the following.Corollary 13.
LetA be a scalar type spectral operator in a complex Banach space. If all weak solutions of (2) are strongly infinite differentiable on (0,∞), then all weak solutions of (1) are strongly infinite differentiable on R.Remark 14.
As follows from Theorem12, all weak solutions of (1) with a scalar type spectral operator A in a complex Banach space can bestrongly infinite differentiable while the operator A isunbounded, e.g., when A is an unboundedself-adjoint operator in a complex Hilbert space (cf. [17, Theorem 7.1]). This fact contrasts the situation when a closed densely defined linear operator A in a complex Banach space generates a strongly continuous group Ttt∈R of bounded linear operators, i.e., the associated abstract Cauchy problem iswell-posed (see Remarks 1.1), in which case even the (left or right) strong differentiability of all weak solutions of (1) at 0 immediately impliesboundedness for A (cf. [3]).


## 6. The Cases of Normal and Self-Adjoint Operators

As an important particular case of Theorem12, we obtainCorollary 15 (the case of a normal operator).
LetA be a normal operator in a complex Hilbert space. Every weak solution of (1) is strongly infinite differentiable on R iff there exist b+>0 and b->0 such that the set σ(A)\Lb-,b+, where (110)Lb-,b+≔λ∈C∣Reλ≤min⁡0,-b-ln⁡ImλorReλ≥max⁡0,b+ln⁡Imλ, is bounded (see Figure 1).Remark 16.
Corollary15 develops the results of paper [17], where similar consideration is given to the strong differentiability of the weak solutions of (2) with a normal operator A in a complex Hilbert space on [0,∞) and (0,∞).From Corollary13, we immediately obtain the following.Corollary 17.
LetA be a normal operator in a complex Hilbert space. If all weak solutions of (2) are strongly infinite differentiable on (0,∞) (cf. [17, Theorem 5.2]), then all weak solutions of (1) are strongly infinite differentiable on R.Considering that, for a self-adjoint operatorA in a complex Hilbert space X,(111)σA⊆R(see, e.g., [10, 11]), we further arrive at the following.Corollary 18 (the case of a self-adjoint operator).
Every weak solution of (1) with a self-adjoint operator A in a complex Hilbert space is strongly infinite differentiable on R.Cf. [17, Theorem 7.1].


## 7. Inherent Smoothness Improvement Effect

As is observed in the proof of the“only if” part of Theorem 12, the opposite to the“if” part’s premise implies that there is a weak solution of (1), which is not strongly differentiable at 0. This renders the case of finite strong differentiability of the weak solutions superfluous and we arrive at the following inherent effect of smoothness improvement.Proposition 19.
LetA be a scalar type spectral operator in a complex Banach space (X,·). If every weak solution of (1) is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.Cf. [1, Proposition 5.1].


## 8. Concluding Remark

Due to thescalar type spectrality of the operator A, Theorem 12 is stated exclusively in terms of the location of itsspectrum in the complex plane, similarly to the celebratedLyapunov stability theorem [19] (cf. [3, Ch. I, Theorem 2.10]), and thus is an intrinsically qualitative statement (cf. [1, 20]).



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*Source: 4168609-2018-07-17.xml*