# On Perturbation of ConvolutedC-Regularized Operator Families

Authors: Fang Li; Jin Liang; Ti-Jun Xiao; Jun Zhang
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Of concern are two classes of convolutedC-regularized operator families: convoluted C-cosine operator families and
convoluted C-semigroups. We obtain new and general multiplicative and additive
perturbation theorems for these convoluted C-regularized operator families. Two examples
are given to illustrate our abstract results.


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## Body

## 1. Introduction

It is well known that the cosine operator families (resp., theC0 semigroups) and the fractionally integrated C-cosine operator families (resp., integrated C-semigroups) are important tools in studying incomplete second-order (resp., first-order) abstract Cauchy problems  (cf., e.g., [1–17]). As an extension of the cosine operator families (resp., the C0 semigroups) as well as the fractionally integrated C-cosine operator families (resp., integrated C-semigroups), the convoluted C-cosine operator families (resp., convoluted C-semigroups) (cf., e.g., [15, 18, 19]) are also good operator families in dealing with ill-posed incomplete second order (resp. first order) abstract Cauchy problems.In last two decades, there are many works on the perturbations on theC-regularized operator families (cf., e.g., [16, 20–24]). In the present paper, we will study the multiplicative and additive perturbation for two classes of convoluted C-regularized operator families: convoluted C-cosine operator families and convoluted C-semigroups, and our purpose is to obtain some new and general perturbation theorems for these convoluted C-regularized operator families and to make the results new even for convoluted n-times integrated C-cosine operator families (resp., convoluted n-times integrated C-semigroups) (n∈ℕ0, where ℕ0 denotes the nonnegative integers).Throughout this paper,ℕ, ℝ, ℂ denote the set of positive integers, the real numbers, and the complex plane, respectively. X denotes a nontrivial complex Banach space, and L(X) denotes the space of bounded linear operators from X into X. In the sequel, we assume that C∈L(X) is an injective operator. C([a,b],X) denotes the space of all continuous functions from [a,b] to X. For a closed linear operator A on X, its domain, range, resolvent set, and the C-resolvent set are denoted by D(A), R(A), ρ(A), and ρc(A), respectively, where ρc(A) is defined by
(1)ρc(A):={λ∈ℂ:R(C)⊂R(λ-A),ρc(A):=λ-Aisinjective}.K∈C([0,∞),ℂ) is an exponentially bounded function, and for β∈ℝ,
(2)ℒ[K(t)](λ)≠0(Reλ>β),
where ℒ[K(t)](λ) is the Laplace transform of K(t) as in the monograph [15]. We define
(3)Θ(t):=∫0t‍K(s)ds,t≥0.Next, we recall some notations and basic results from [15, 19] about the convoluted C-cosine operator families and convoluted C-semigroups.The following definition is the convoluted version of [15, Chapter 1, Definition 4.1].Definition 1.
Letω≥0 and (ω2,∞)⊂ρc(A). Let {CK(t)}t≥0(CK(t)∈L(X),t≥0) be a strongly continuous operator family such that
(4)∥CK(t)∥≤Meωt,t≥0,
for some M>0, and
(5)λ(λ2-A)-1Cx=1ℒ[K(t)](λ)∫0∞e-λtCK(t)xdt,Reλ>max(ω,β),x∈X.
Then, A is called a subgenerator of the exponentially bounded K-convoluted C-cosine operator family {CK(t)}t≥0. Moreover, the operator A¯:=C-1AC is called the generator of the {CK(t)}t≥0.Proposition 2.
LetA be a closed operator and {CK(t)}t≥0 a strongly continuous, exponentially bounded operator family. Then A is the subgenerator of a K-convoluted C-cosine operator family {CK(t)}t≥0 if and only if(1)
CK(t)C=CCK(t),  t≥0;(2)
CK(t)A⊂ACK(t),  t≥0, and
(6)A∫0t∫0sCK(σ)xdσds=CK(t)x-Θ(t)Cx,t≥0,x∈X.Remark 3.
IfA is the subgenerator of a K-convoluted C-cosine operator family, then CA⊆AC.Definition 4.
Let0≤ω<∞ and (ω,∞)⊂ρc(A). Let {TK(t)}t≥0 be a strongly continuous operator family such that
(7)∥TK(t)∥≤Meωt,t≥0,
for some M>0, and
(8)(λ-A)-1Cx=1ℒ[K(t)](λ)∫0∞e-λtTK(t)xdt,Reλ>max{ω,β},x∈X.
Then, A is called a subgenerator of an exponentially bounded K-convoluted C-semigroup {TK(t)}t≥0. Moreover, the operator A¯:=C-1AC is called the generator of the {TK(t)}t≥0.Proposition 5.
LetA be a closed operator, and {TK(t)}t≥0 a strongly continuous, exponentially bounded operator family. Then, A is the subgenerator of a K-convoluted C-semigroup {TK(t)}t≥0 if and only if(1)
TK(t)C=CTK(t),  t≥0;(2)
TK(t)A⊂ATK(t),  t≥0, and
(9)A∫0tTK(s)xds=TK(t)x-Θ(t)Cx,t≥0,x∈X.Remark 6.
From [15], we know that the C-cosine operator families (resp., C-semigroups) are exactly the 0-times integrated C-cosine operator families (resp., the 0-times integrated C-semigroups). Let Γ(·) be the well-known Gamma function, and
(10)K(t)=tα-1Γ(α).
Then, by Propositions 2 and 5, we get results for the α-times integrated C-cosine operator families (resp., α-times integrated C-semigroups) as well as C-cosine operator families (resp., C-semigroups). For more information on various C operator families, we refer the reader to, for example, [3, 6–8, 14, 15, 17, 22] and references therein.


## 2. Multiplicative Perturbation Theorems

Lemma 7.
Suppose thatA is a subgenerator of an exponentially bounded K-convoluted C-cosine operator family on X. If ρ(A)≠∅, then C-1AC=A.Proof.
For anyλ0∈ρ(A) and x∈D(C-1AC), let
(11)y=λ0x-C-1ACx.
Then,
(12)(λ0-A)-1C=C(λ0-A)-1,Cx=(λ0-A)-1Cy=C(λ0-A)-1y.
Therefore,
(13)x=(λ0-A)-1y∈D(A).
This means that C-1AC⊆A. Thus, by Remark 3, we see that C-1AC=A.Theorem 8.
LetA be a closed linear operator on X and ℛ∈L(X). Assume that there exists an injective operator C on X satisfying CA⊆AC, ℛC=Cℛ. Then, the following statements hold.(1)
IfℛA subgenerates an exponentially bounded K-convoluted C-cosine operator family on X, then Aℛ subgenerates an exponentially bounded K-convoluted C-cosine operator family on X.(2)
IfAℛ subgenerates an exponentially bounded K-convoluted C-cosine operator family on X and ρ(ℛA)≠∅, then ℛA generates an exponentially bounded K-convoluted C-cosine operator family on X.Proof.
(
1
) Assume that ℛA subgenerates an exponentially bounded K-convoluted C-cosine operator family {CK(t)}t≥0 on X.
In this case, it is easy to see that for anyt≥0, the operator
(14)x↦A∫0tSK(s)ℛxds
is bounded, since
(15)∫0tSK(s)ℛxds∈D(ℛA),
where SK(t)=∫0t‍CK(s)ds. Now, for each t≥0, we define a bounded linear operator as follows:
(16)C^K(t)x=Θ(t)Cx+A∫0tSK(s)ℛxds.
Clearly, the graph norms of ℛA and A are equivalent. Therefore, noting that ℛA subgenerates an exponentially bounded K-convoluted C-cosine operator family {CK(t)}t≥0 on X, we obtain, for every t1,t2≥0, and x∈X, that there exists a constant M1 such that
(17)∥C^K(t1)x-C^K(t2)x∥≤∥Θ(t1)Cx-Θ(t2)Cx∥+∥A(∫0t1SK(s)ℛxds-∫0t2SK(s)ℛxds)∥≤∥Θ(t1)Cx-Θ(t2)Cx∥+M1(∥ℛA(∫0t1SK(s)ℛxds-∫0t2SK(s)ℛxds)∥+M1+∥∫0t1SK(s)ℛxds-∫0t2SK(s)ℛxds∥)=(M1+1)∥Θ(t1)Cx-Θ(t2)Cx∥+M1∥CK(t1)x-CK(t2)x∥+∥∫0t1SK(s)ℛxds+-∫0t2SK(s)ℛxds∥⟶0,ast1⟶t2.
Hence, C^K(·) is strongly continuous.
Similarly, we can prove thatC^K(·) is exponentially bounded; that is, there exists a constant M^>0 such that
(18)∥C^K(t)∥≤M^eωt,t≥0.
As in the monograph [15], we write
(19)ℒ[C^K(t)](λ)x=∫0∞e-λtC^K(t)xdt,forReλ>max(ω,β),x∈X.
Then, by (16), we have
(20)ℒ[C^K(t)](λ)x=ℒ[K(t)](λ)λCx+Aℒ[K(t)](λ)λ(λ2-ℛA)-1Cℛx.
Hence,
(21)ℛℒ[C^K(t)](λ)x=ℒ[K(t)](λ)λ×C[ℛx+ℛA(λ2-ℛA)-1Cℛx]=λℒ[K(t)](λ)(λ2-ℛA)-1Cℛx∈D(A).
Furthermore,
(22)(λ2-Aℛ)ℒ[C^K(t)](λ)x=λ2ℒ[C^K(t)](λ)x-λℒ[K(t)](λ)A(λ2-ℛA)-1Cℛx=λℒ[K(t)](λ)Cx.
On the other hand, for each x∈D(Aℛ), Reλ>max(ω,β), we obtain
(23)ℒ[K(t)](λ)λ[C+A(λ2-ℛA)-1Cℛ](λ2-Aℛ)x=λℒ[K(t)](λ)Cx.
Therefore,
(24)λ(λ2-Aℛ)-1C=1λ[I+A(λ2-ℛA)-1ℛ]C.
It follows from (20) that
(25)ℒ[C^K(t)](λ)x=λℒ[K(t)](λ)(λ2-Aℛ)-1Cx.
Thus, by Definition 1, we know that Aℛ subgenerates an exponentially bounded K-convoluted C-cosine operator family on X.
(2) Assume thatAℛ subgenerates an exponentially bounded K-convoluted C-cosine operator family on X and ρ(ℛA)≠∅, and let
(26)λ0∈ρ(ℛA),E=(λ0-ℛA)ℛ,F=A(λ0-ℛA)-1.
It is not hard to see that E is closed operator on X and
(27)F∈L(X),CE⊆EC,FC=CF.
Since FE=Aℛ subgenerates an exponentially bounded K-convoluted C-cosine operator family on X, we know from (1) that the operator EF=ℛA subgenerates an exponentially bounded K-convoluted C-cosine operator family on X.
Noting thatρ(ℛA)≠∅ and in view of Lemma 7, we see that ℛA generates an exponentially bounded K-convoluted C-cosine operator family on X.Theorem 9.
LetA be a subgenerator of an exponentially bounded K-convoluted C-cosine operator family {CK(t)}t≥0 on X,
(28)SK(t)=∫0tCK(s)ds,t≥0,B∈L(X), and R(B)⊂R(C). Suppose that(H1)
there exists an operatorℱ:X→X such that
(29)ℱSK(t)x:=GK(t)x∈C([0,∞),X)
is Laplace transformable, and
(30)ℒ(GK)(λ)=(λ2-A)-1Cx,x∈X;(H2)
for anyΦ∈C([0,∞),X), ∫0t‍GK(t-s)C-1BΦ(s)ds∈D(A), and
(31)∥A∫0tGK(t-s)C-1BΦ(s)ds∥≤M~∫0teω(t-s)∥Φ(s)∥ds,t≥0,whereM~ is a constant;(H3)
there exists an injective operatorC1∈L(X) such that R(C1)⊂R(C) and C1A(I+B)⊂A(I+B)C1. Then,(1)
A(I+B) subgenerates an exponentially bounded K-convoluted C1-cosine operator family,(2)
ifρ(A)≠∅, then A(I+B) generates an exponentially bounded K-convoluted C1-cosine operator family;(3)
ifρ((I+B)A)≠∅ and BC1=C1B, C1A⊆AC1, then (I+B)A generates an exponentially bounded K-convoluted C1-cosine operator family on X.Proof.
(
1
) For each x∈X, t≥0, define
(32)C¯0(t)x=CK(t)x,C¯n(t)x=A∫0tGK(t-s)C-1BC¯n-1(s)xds,n=1,2,….
Then, the operator family {C¯n(t)}t≥0 has the following properties:(i)
for anyx∈X, C¯n(t)x∈C([0,∞),X);(ii)
∥C¯n(t)∥≤(MM~ntn/n!)eωt,t≥0,∀n∈ℕ0. Therefore, the following series
(33)∑n=0∞C¯n(t)C-1C1,t≥0,
is uniformly convergent on every compact interval in t, and we set
(34)h(t)=∑n=0∞C¯n(t)C-1C1,t≥0.
Clearly,
(35)∥h(t)∥≤M1e(ω+M~)t,t≥0,
where M1=M∥C-1C1∥, and
(36)t⟶h(t)xiscontinuouson[0,∞)foranyx∈X.
Moreover,
(37)h(t)x=CK(t)C-1C1x+A∫0t‍GK(t-s)C-1Bh(s)xds,x∈X,t≥0.
As in the monograph [15], we write, for sufficiently large λ,
(38)ℒ[h(t)](λ)x=∫0∞e-λth(t)xdt,x∈X.
Thus, by (5), we have
(39)ℒ[h(t)](λ)x=λℒ[K(t)](λ)(λ2-A)-1C1x+A(λ2-A)-1Bℒ[h(t)](λ)x,x∈X.
This implies that
(40)R((I+B)ℒ[h(t)](λ))⊆D(A),(λ2-A(I+B))ℒ[h(t)](λ)x=λ2ℒ[h(t)](λ)x-λℒ[K(t)](λ)A(λ2-A)-1C1x-λ2A(λ2-A)-1Bℒ[h(t)](λ)x=λℒ[K(t)](λ)C1x,x∈X.
Let
(41)U(t)x=A∫0tGK(t-s)C-1Bxds,x∈X,t≥0.
Then, for large λ, we have
(42)∥λℒ[U(t)](λ)∥=∥λ∫0∞e-λtU(t)dt∥≤M~λ-ω.
So, for sufficiently large λ,
(43)∥λℒ[K(t)](λ)∥=∥A(λ2-A)-1B∥<1.
This means that the operator I-A(λ2-A)-1B is invertible.
On the other hand, sinceλ2-A and I-A(λ2-A)-1B are injective, and
(44)(λ2-A)(I-A(λ2-A)-1B)x=(λ2-A(I+B))x,x∈D(A(I+B)),
we infer that λ2-A(I+B) is injective. This together with (40) implies that
(45)λ(λ2-A(I+B))-1C1x=1ℒ[K(t)](λ)∫0∞e-λth(t)xdt.
By Definition 1, we know that A(I+B) subgenerates an exponentially bounded K-convoluted C1-cosine operator family on X.
(
2
) By the proof of (1), we see that the operator I-A(λ2-A)-1B is invertible, and ρ(A)≠∅ implies that
(46)ρ(A(I+B))≠∅.
In view of Lemma 7, we get
(47)C1-1A(I+B)C1=A(I+B).
(
3
) By virtue of Theorem 8 (2), we have the conclusion.Remark 10.
(
1
) It is easy to see that if we take
(48)ℱSK(t)x:=(ℒ-1(1ℒ[K(t)](λ))*SK)(t)x,
then (H1) is satisfied.
(2) In Theorem9, if we take
(49)K(t)=tn-1Γ(n),ℱ:=dndtn,n∈ℕ,
then we obtain the perturbations for n-times integrated C-cosine operator families.
(3) In Theorem9, if we take
(50)K(t)≡1t(t≠0)
and ℱ:=I, then we have the multiplicative perturbations on the exponentially bounded C-cosine operator families.
By Theorem9, we can immediately deduce the following theorem on K-convoluted C-semigroups.Theorem 11.
LetA be a subgenerator of an exponentially bounded K-convoluted C-semigroup {TK(t)}t≥0 on X, B∈L(X) and R(B)⊂R(C). Suppose that(H1)
there exists an operatorℱ:X→X such that
(51)ℱTK(t)x:=HK(t)x∈C([0,∞),X)is Laplace transformable, and(52)ℒ(HK)(λ)=(λ-A)-1Cx,x∈X;(H2)
for anyΦ∈C([0,∞),X), ∫0t‍HK(t-s)C-1BΦ(s)ds∈D(A), and
(53)∥A∫0tHK(t-s)C-1BΦ(s)ds∥≤M¯∫0teω(t-s)∥Φ(s)∥ds,t≥0,whereM¯ is a constant;(H3)
there exists an injective operatorC1∈L(X) such that R(C1)⊂R(C) and C1A(I+B)⊂A(I+B)C1.Then,(1)
A(I+B) subgenerates an exponentially bounded K-convoluted C1-semigroup on X;(2)
ifρ(A)≠∅, then A(I+B) generates an exponentially bounded K-convoluted C1-semigroup on X.(3)
ifρ((I+B)A)≠∅, then (I+B)A generates an exponentially bounded K-convoluted C1-semigroup on X.Remark 12.
(1) In Theorem11, if we take
(54)K(t):=tn-1Γ(n),ℱ:=dndtn,n∈ℕ,
then we obtain the perturbations for n-times integrated C-semigroups.
(2) In Theorem11, if we take
(55)K(t):=1t(t≠0)
and ℱ:=I, then we have the multiplicative perturbations on the exponentially bounded C-semigroups.


## 3. Additive Perturbation Theorem

Theorem 13.
LetB∈L(X), R(B)⊂R(C), and there exists an injective operator C1∈L(X) such that R(C1)⊂R(C) and C1(A+B)⊂(A+B)C1.(i)
Suppose thatA is a subgenerator of an exponentially bounded K-convoluted C-cosine operator family {CK(t)}t≥0 on X. If there exists an operator ℱ:X→X such that
(56)ℱCK(t)x:=GK(t)x∈C([0,∞),X)
is Laplace transformable, and
(57)ℒ(GK)(λ)=(λ2-A)-1Cx,x∈X,
then A+B subgenerates an exponentially bounded K-convoluted C1-cosine operator family {C^K(t)}t≥0 on X, where
(58)C^K(t)x=CK(t)C-1C1x+∫0tSK(t-s)C-1BC^K(t)xds,t≥0,x∈X,SK(t)x=∫0tCK(s)xds,t≥0,x∈X.(ii)
Suppose thatA is a subgenerator of an exponentially bounded K-convoluted C-semigroup {TK(t)}t≥0 on X. If there exists an operator ℱ:X→X such that
(59)ℱTK(t)x:=HK(t)x∈C([0,∞),X)
is Laplace transformable, and
(60)ℒ(HK)(λ)=(λ-A)-1Cx,x∈X,
then A+B subgenerates an exponentially bounded K-convoluted C1-semigroup {T^K(t)}t≥0 on X, where
(61)T^K(t)x=TK(t)C-1C1x+∫0tTK(t-s)C-1BT^K(s)xds,t≥0,x∈X.Proof.
Replacing (37) with the following equality:
(62)h(t)x=CK(t)C-1C1x+∫0tGK(t-s)C-1Bh(s)xds,x∈X,t≥0,
and by the arguments similar to those in the proof of Theorem 9, we can prove (i).
Point (ii) can also be deduced by a similar way.Remark 14.
In Theorem13, if we take
(63)K(t)=tn-1Γ(n),ℱ:=dndtn,n∈ℕ0,
then we obtain an additive perturbation theorem for the exponentially bounded n-times integrated C1-cosine operator families (resp., n-times integrated C1-semigroups) as well as C1-cosine operator families (resp., 0-times integrated C1-semigroup).


## 4. Examples

Example 1.
Let(64)X:=C0(ℝ)⊕C0(ℝ)⊕C0(ℝ),A(f,g,h)(·):=(f′,g′,(χ[0,∞)-χ(-∞,0])h),
where
(65)(f,g,h)∈D(A)={(f,g,h)∈X:f′∈C0(ℝ),g′∈C0(ℝ),h(0)=0},C(f,g,h):=(f,g,sin(·)h(·)),f,g,h∈C0(ℝ).
As in [22, Examples 8.1 and 8.2], we can prove that A is the generator of an exponentially bounded once integrated C-semigroup ([15]).
Define(66)B(f,g,h)(t)=(e-tcost∫0tf(s)ds,e-2tcos2t×∫0tg(s)ds,te-6tsint·h(t)),
for every t∈ℝ, and f,g,h∈C0(ℝ). Then, we can simply verify B∈L(X), R(B)⊂C(D(A)), and
(67)BC(f,g,h)=CB(f,g,h),(f,g,h)∈X.
Therefore, taking
(68)K(t)≡1,ℱ:=ddt
and using Remark 12 (1), we know that A(I+B) subgenerates an exponentially bounded once integrated C-semigroup on X.Example 2.
LetX1=L∞(ℝ), X2=L2(ℝ),
(69)A1=d2dξ2,D(A1)=W2,∞(ℝ),A2=d2dξ2,D(A2)=H2(ℝ).
It follows from [15] that A1 generates an exponentially bounded C1-cosine operator family C1(·) on X1, where C1=(1-d2/dξ2)-1. Moreover, it is well known that A2 generates a strongly continuous cosine operator family C2(·) on X2.
Let(70)b1(·)∈W4,∞(ℝ),b2(·)∈H2(ℝ),
and define B1:X2→X1, B2:X1→X2 as follows:
(71)(B1ϕ)(ξ)=b1(ξ)∫01ϕ(σ)dσ,(B2ϕ)(ξ)=b2(ξ)∫01ϕ(σ)dσ.
Set X=X1×X2,
(72)A=(A100A2),D(A):=D(A1)×D(A2),B=(0B1B20),D(B):=X.
Clearly, ρ(A)≠∅ and D(A1)=R(C1). Take
(73)λ0∈ρ(A),C=(λ0-A)-1.
Then, A generates an exponentially bounded C-cosine operator family C(·) on X, where
(74)C(t)=(C1(t)C1-1(λ0-A1)-100C2(t)(λ0-A2)-1).
Hence,
(75)S(t)=(S1(t)C1-1(λ0-A1)-100S2(t)(λ0-A2)-1),
where
(76)S(t):=∫0t‍C(s)ds,S1(t):=∫0tC1(s)ds,(77)S2(t):=∫0tC2(s)ds.
Therefore, we have, for each f=(f1f2)∈C([0,∞),X),
(78)∫0tS(t-s)C-1Bf(s)ds=(A1∫0t‍S1(t-s)C1-1B1f2(s)dsA2∫0t‍S2(t-s)B2f1(s)ds).
Since
(79)R(B1)⊂D(A1C1-1),R(B2)⊂D(A2),
we see that there exist M, ω>0 such that
(80)∥∫0tS(t-s)C-1Bf(s)ds∥≤M∫0teω(t-s)∥f(s)∥ds,t≥0.
Consequently, if there exists an injective operator C~∈L(X) such that R(C~)⊂R(C) and C~(A+B)⊂(A+B)C~, then taking
(81)K(t)≡1t,ℱ:=I
and using Remark 14, we know that A+B subgenerates a C~-cosine operator family on X.
Moreover, it is not hard to see that there existM^, ω>0 such that
(82)∥A∫0tS(t-s)C-1Bf(s)ds∥≤M^∫0teω(t-s)∥f(s)∥ds,t≥0.
Hence, if there exists an injective operator C^∈L(X) such that R(C^)⊂R(C) and C^A(I+B)⊂A(I+B)C^, then by Remark 10 (3) (ρ(A)≠∅), we know that A(I+B) generates a C^-cosine operator family on X.



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