On Perturbations of Generators of -Semigroups

Abstract

We present a perturbation result for generators of -semigroups which can be considered as an operator theoretic version of the Weiss-Staffans perturbation theorem for abstract linear systems. The results are illustrated by applications to the Desch-Schappacher and the Miyadera-Voigt perturbation theorems and to unbounded perturbations of the boundary conditions of a generator.

1. Introduction

In his classic [1] “Perturbation Theory for Linear Operators”, Kato addresses, among others, the following general problem.

Given (unbounded) operators and on a Banach space , how should one define their “sum” and which properties of are preserved under the perturbation by ?

In the present paper we study this problem in the context of operator semigroups. Given the generator of a -semigroup on , for which operators is the (in a suitable way defined) sum again a generator?

Numerous results are known in this field (see, e.g., [2, Sections III.1–3 and related notes]), but no unifying and general theory is yet available.

Our aim is to go a step towards a more systematic perturbation theory for such generators. To this end we choose the following setting. For the generator with domain consider perturbations where is the extrapolated space associated with (see [2, Section II.5.a]). The sum is then defined as ; that is, For which remains a generator on ? The bounded perturbation theorem ([2, Section III.1]), the Desch-Schappacher ([2, Section III.3.a]), and the Miyadera-Voigt theorems ([2, Section III.3.c]) give some well-known answers in these cases.

It seems that the Weiss-Staffans theorem on the well-posedness of perturbed linear systems (cf. [3, Theorems 6.1 and 7.2] and [4, Sections 7.1 and 7.4]) is a general result in this direction. In the present paper we formulate and prove this result in a purely operator theoretic way avoiding, in particular, notions like abstract linear systems and Lebesgue- or Yosida-extensions.

More precisely (here we use the notation of Weiss, cf. [3]), the classical Weiss-Staffans theorem starts from an abstract linear system, that is, a quadruple of operator families verifying a set of functional equations (for the precise definition see [3, Definition 5.1]). It then shows that for an admissible feedback operator (cf. [3, Definition 3.5]) there exists a unique corresponding closed-loop system . Moreover, it relates the generating operators and of these two systems. Since here and are -semigroups with generators and , respectively, this result implicitly contains a perturbation theorem for generators of -semigroups.

However, to apply this theorem to a perturbed operator as appearing in (2) one first has to construct an abstract linear system with appropriate generating operators and a suitable admissible feedback operator incorporating the unperturbed generator and the perturbation . This makes it quite cumbersome to formulate and to apply the Weiss-Staffans theorem as a perturbation result for generators.

For this reason we start directly from a triple of operators and then give conditions in terms of the semigroup generated by and the operators and implying that for generates a -semigroup. Even though in our approach it is not necessary, it is nevertheless helpful to interpret the perturbed generator as the state operator of a control system with feedback in order to give some motivation for the various definitions of “admissibility.” For this reason in the sequel we use some common terminology from control theory.

More precisely, choose two Banach spaces and called state- and observation-/control space, respectively. (We assume that the observation and control spaces coincide. This is no restriction of generality and somewhat simplifies the presentation.) On these spaces consider the following operators:(i), called the state operator (of the unperturbed system);(ii), called the control operator;(iii), called the observation operator,where is the generator of a -semigroup on . Moreover, is a Banach space such that where “” denotes a continuous linear injection and is the domain equipped with the graph norm. Then consider the linear control system The solution of is formally given by the variation of parameters formula Closing this system by putting , one formally obtains the perturbed abstract Cauchy problem which is well-posed in if and only if for is a generator on (cf. [2, Section II.6]).

Before elaborating this idea, we give a short summary of this paper.

Section 2 is dedicated to the notions of admissibility for control, observation, feedback, and pairs of operators. In Section 3 we state and prove the main results, that is, Theorems 10 and 14. In Section 4 we show how the Desch-Schappacher and Miyadera-Voigt theorems easily follow from Theorem 14 and give an application to the perturbation of the boundary condition of a generator in the spirit of Greiner [5].

2. Admissibility

Being only interested in the generator property of for some perturbation , we can in the sequel assume without loss of generality that the growth bound and hence Taking in the system and considering the initial value it is natural to ask that for every control function one obtains a state for some/all . Hence formula (5) is leading to the following definition (cf. [6, Definition 4.1], see also [7]).

Definition 1. The control operator is called -admissible for some if there exists such that

Note that (8) becomes less restrictive for growing .

Remark 2. The range condition (8) in the previous definition means that the operator given by has range . Since obviously , the closed graph theorem implies that for admissible the controllability map belongs to . On the other hand, using integration by parts, it follows that for every Since is dense in , this shows that the range condition (8) is equivalent to the existence of some such that

Next, consider with . Then it is reasonable to ask that every initial value gives rise to an observation for some/all which also depends continuously on . This yields the following definition (cf. [8, Definition 6.1], see also [7]).

Definition 3. The observation operator is called -admissible for some if there exist and such that

Note that (12) becomes more restrictive for growing .

Remark 4. The norm condition (12) in the previous definition combined with the denseness of implies that there exists an observability map satisfying such that

Finally, consider the system with (possibly nonzero) -admissible control and observation operators and . The following compatibility condition is needed to proceed (cf. [9, Section II.A]). For more information and several related conditions see [10, Theorem 5.8] and [4, Definition 5.1.1]. Recall that .

Definition 5. The triple (or the system ) is called compatible if for some we have

If the inclusion (14) holds for some , then it holds for all by the resolvent identity. Moreover, the closed graph theorem implies that the operator Consider now a compatible control system with initial value . Then the input-output map of which maps a control to the corresponding observation by (5) is formally given by Of course, the right-hand side does not in general make sense for arbitrary since the integral might give values . However, if then integrating by parts twice and using (14) one obtains At this point it is reasonable to ask that the input-output map is continuous. This gives rise to the following definition.

Definition 6. The pair (or the system ) is called jointly -admissible for some if is a -admissible control operator and is a -admissible observation operator and there exist and such that

Remark 7. If is jointly -admissible, then there exists a bounded input-output map We need a further definition.

Definition 8. An operator is called a -admissible feedback operator for some if there exists such that is invertible.

Note that is admissible if . For further reference we summarize some of the previous notions in a single notation.

Definition 9. Let be the generator of a -semigroup on a Banach space , and for a Banach space satisfying . Then is called a Weiss-Staffans perturbation for if for some the following hold:(i) is a compatible triple;(ii) is a -admissible control operator;(iii) is a -admissible observation operator;(iv) is a -admissible pair;(v) is a -admissible feedback operator.

3. The Weiss-Staffans Perturbation Theorem

In this section we state and prove the main results of this paper. These results can be considered as purely operator theoretic versions of perturbation theorems for abstract linear systems due to Weiss [3, Theorems 6.1 and 7.2 (1994)] in the Hilbert space case and Staffans [4, Theorems 7.1.2 and 7.4.5 (2005)] for Banach spaces. In particular, our approach avoids the use of the notions of abstract linear systems and Lebesgue extensions which are not needed if one is only interested in generators. For related results see also [11] and [12, Theorems 4.2 and 4.3].

Theorem 10. Assume that is a Weiss-Staffans perturbation for . This means that there exist , and such that is  given  by  (20). Then generates a -semigroup on the Banach space . Moreover, the perturbed semigroup verifies the variation of parameters formula

For the proof we extend the controllability-, observability-, and input-output maps introduced in Remarks 2, 4, and 7 on as follows. Recall that by assumption .

Lemma 11. Let be compatible and jointly -admissible for some . Then there exist(i)a strongly continuous, uniformly bounded family ;(ii)a bounded operator ;(iii)a bounded operator ,such that

Proof. The assertion for was proved in [13, Corollary 3.16]. The assertion for was shown in [13, Lemma 3.9]. Finally, the assertion for follows from [13, Remark 3.23].

For we indicate in the sequel the controllability-, observability-, and input-output maps associated with the triple with the superscript “”, for example, Lemma 12 gives a condition such that the invertibility of (see condition of Theorem 10) implies the one of for sufficiently large.

Lemma 12. Let the assumptions of Theorem 10 be satisfied. If for holds, then .

Proof. Inspired by [14, (2.6)] and the proof of [15, Proposition 2.1] consider for the surjective isometry (denote by the transposed vector of a vector ) where , , and .
Then is isometrically isomorphic to the matrix Since by assumption is invertible, is invertible as well and where we put . By Lemma A.1 applied to one obtains the estimate This shows that remains bounded as if (26) holds for .
If the estimate (26) only holds for some , consider the triple . Let be the multiplication operator defined by Then is invertible with inverse and a simple computation shows that By similarity this implies that . Hence, repeating the above reasoning for one obtains from (30) that remains bounded as if Since by (32) one has the estimates (33) and (26) are equivalent. Summing up this shows that (26) implies that Using this fact we finally show that . Observe first that for some implies that for every . Since is injective for every , this gives that ; that is, is injective.
To show surjectivity fix some and define for that is, is the unique solution in of the equation However, for one has ; hence also solves (37). This implies that Thus one can define Since by (35) it follows that for all , Fatou’s lemma implies that . Moreover, by construction which implies . Since was arbitrary, this shows that is surjective. Hence is bijective and therefore as claimed.

Next we show that the invertibility of implies for sufficiently large the invertibility of the “transfer function” of the system with feedback . In the following the Laplace transform of a function is denoted by

Lemma 13. Assume that for some . Then for all satisfying and

Proof. Assume first that . Then it is well known that is shift invariant (cf. [16]); that is, commutes with the right shift. Then also is shift invariant and by [16, Theorem  2.3] and [13, Lemma 3.19] one obtains for Let . Then clearly the right shift also commutes with ; that is, this operator is shift invariant as well. Hence again by [16, Theorem  2.3] there exists such that Summing up one obtains for all Taking for some , this implies Hence .
If , then by the same reasoning applied to one obtains that Clearly this implies our claim in case and the proof is complete.

We are now well prepared to prove the main result of this section.

Proof of Theorem 10. The idea of the proof is to define an operator family and then to verify that it is a -semigroup with generator .
To this end, assume that the condition (26) in Lemma 12 holds for . Then is invertible and one can define for Since and are both strongly continuous and uniformly bounded, the same holds for . We proceed and compute the Laplace transform of for . Since the convolution theorem for the Laplace transform (or [13, Lemma 3.12]) and Lemma 13 imply for every and We now show that . First note that by the compatibility condition (14) one has Moreover, This implies that is a right inverse and . To show that it is also a left inverse take . Then we obtain and hence it follows that as claimed. Summing up we showed that is a strongly continuous family with Laplace transform . By [17, Theorem 3.1.7] this implies that is a -semigroup with generator .
To verify the variation of parameters formula (23) one first notes that by Lemma 13 and the explicit representation of in (50) one has for all and that By the uniqueness of the Laplace transform this implies that and the assertion follows from the definition of in (49).
Now assume that (26) only holds for some . Then repeating the same reasoning for the triple one concludes as before that is a generator. Clearly this implies that   generates a strongly continuous semigroup . Moreover, one obtains that the pair of rescaled semigroups and verify the variation of parameters formula (23) which implies that this formula holds for the pair and as well.

As already remarked in the introduction, with increasing the -admissibility of the control and observation operator becomes weaker and stronger, respectively.

Assuming that the input-output map maps to for some (the main cases we have in mind are and ) satisfying , one can drop the invertibility condition in Theorem 10 (and sometimes even the compatibility condition (14), cf. Remark 15).

Theorem 14. Assume that conditions (i)–(iii) in Theorem 10 are satisfied. Moreover, suppose there exist with , , and such that Then given by (22) generates a -semigroup on the Banach space verifying the variation of parameters formula (23).

Proof. By Theorem 10 it suffices to show that for some . By assumption the operator has a bounded extension for every . We distinguish 2 cases and use in both of them Jensen’s inequality as follows: for and .
Case . Then belongs to with norm (denote the norm of a bounded linear operator by )  . This implies, by (58) for and , that
Case . In this case, with norm . This implies, by (58) for and , that Hence in both cases, considering , one concludes that there exists such that which implies .

Remark 15. Assume as in Theorem 14 that with and . If there exist and a dense subspace such that for every (i) for almost all ;(ii)the map is in ;(iii)there exists such that then also the compatibility condition (14) is satisfied.
To verify this assertion define as in (57) with replaced by the space . By assumption, has a unique bounded extension . As above take . Then, by Hölder’s inequality (or (58) for and ), one obtains for every (define for all where is a scalar function; moreover, by denote the constant one function on the interval ) By [10, Theorem 5.8] in the Hilbert space case or [4, Theorems 5.6.4 and 5.6.5] in the general case this convergence implies the compatibility condition (14).

4. Applications

We now give some applications of our abstract results. First we show that Theorem 14 can be considered as a simultaneous generalization of the Desch-Schappacher and the Miyadera-Voigt perturbation theorems. Moreover, we generalize a result of Greiner concerning the perturbation of the boundary conditions of a generator.

4.1. The Desch-Schappacher Perturbation Theorem

The following result was proved in [18, Theorem 5, Proposition 8]; see also [2, Corollary III.3.4] and [19, Corollary 5.5.1].

Theorem 16 (see [18]). Assume that for there exist , , and such that Then given by is the generator of a -semigroup on .

Remark that one could consider the condition (63) also for or . However, in this case one needs an additional norm estimate to ensure that condition (v) in Theorem 10 is satisfied (cf. [2, Corollary III.3.3.]). Moreover, note that in a certain sense the Desch-Schappacher theorem depends only on the range but not on the “size” of the perturbation . In particular, if satisfies the assumption of Theorem 16, then also satisfies it for every .

Proof of Theorem 16. Let and . Then by assumption is a -admissible control operator and conditions (i)–(iii) in Theorem 10 are clearly satisfied. We will prove that (ii) implies condition from Theorem 14. To this end we first verify that the function is continuous for every . For such define by that is, is just the right translation of by . Then and using Remark 2 one obtains from that for where the last step follows from the strong continuity of the nilpotent right translation semigroup on . Next define the operator The operator is well-defined. Moreover, the estimate shows that for all . Choosing this implies condition and hence the proof is complete.

Remark 17. The proofs of Theorems 14 and 16 imply the following: if is a -admissible control operator for some then for every bounded the triple is compatible and jointly -admissible. Moreover, in this case every is a -admissible feedback operator for the system .

4.2. The Miyadera-Voigt Perturbation Theorem

As another application we consider the following version of the Miyadera-Voigt perturbation theorem (cf. [20, 21], see also [2, Corollary III.3.16] and [19, Theorem 5.4.2]).

Theorem 18 (see [20, 21]). Assume that for there exist , , and such that Then given by is the generator of a -semigroup on .

Observe that one could consider condition (70) also for . However, in this case one needs to ensure that condition (v) in Theorem 10 is satisfied (cf. [2, Corollary III.3.16]). Moreover, note that in a certain sense the Miyadera-Voigt Theorem 18 (for ) depends only on the domain but not on the “size” of the perturbation . In particular, if satisfies the assumption of Theorem 18, then also satisfies it for every .

Proof of Theorem 18. Let , , and . Then, by assumption, is a -admissible observation operator and conditions (i)–(iii) in Theorem 10 are clearly satisfied. We will show that condition (iii) implies condition from Theorem 14. To this end fix and . Then for one obtains Using (72), condition (iii), the triangle and Hölder’s inequality for and Let now be a step function where the intervals are pairwise disjoint and for . Then from (73) one obtains Since the step functions having values in are dense in , this implies condition for and . This completes the proof.

Remark 19. The proofs of Theorems 14 and 18 imply the following: if is a -admissible observation operator for some then for every bounded the triple is compatible and jointly -admissible. Moreover, if then in addition every is a -admissible feedback operator for the system .

4.3. Perturbing the Boundary Conditions of a Generator

In this section we show how Theorem 10 can be used to generalize significantly the approach by Greiner in [5] to perturbations of boundary conditions of a generator. To explain the general setup we consider the following:(i)two Banach spaces (in this section denote the elements of by instead of )    and , the latter called “boundary space”;(ii)a closed, densely defined “maximal” operator (“maximal” concerns the size of the domain, e.g., a differential operator without boundary conditions) ;(iii)the Banach space where is the graph norm;(iv)two “boundary” operators .Then define two restrictions by In many applications , , and are function spaces and is a “trace-type” operator which restricts a function in to (a part of) the boundary of its domain. Hence one can consider with boundary condition as a perturbation of the operator with abstract “Dirichlet type” boundary condition .

In order to treat this setup within our framework we make the following assumptions:(i)the operator generates a strongly continuous semigroup on ;(ii)the boundary operator is surjective.Under these assumptions the following lemma, shown by Greiner [5, Lemma 1.2], is the key to write as a Weiss-Staffans perturbation of .

Lemma 20. Let the above assumptions (i) and (ii) be satisfied. Then for each the operator is invertible and is bounded.

Using this so-called Dirichlet operator one obtains the following representation of where for simplicity we assume that is invertible.

Lemma 21. If , then that is, for  , , and

Proof. Denote the operator on the right-hand side of (76) by . Then Moreover, for we have as claimed.

We mention that in [5, Theorem  2.1] the operator is bounded and the assumptions imply that is a -admissible control operator. Hence in this case is a generator by the Desch-Schappacher theorem.

By using Theorem 10 one can now deal also with unbounded .

Corollary 22. Assume that for some the pair is jointly -admissible and that is a -admissible feedback operator for . Then is the generator of a -semigroup on .

Proof. One only has to show the compatibility condition (14). This, however, immediately follows from

Remark 23. We note that in [22, Theorem 4.1] the authors study a similar problem in the context of regular linear systems.

As a simple but typical example for the previous corollary consider the space and the first derivative with domain (c.f. [5, Example 1.1.(c)]). As boundary space choose , as boundary operator the point evaluation and as perturbation some . This gives rise to the differential operators with domains Then clearly the assumptions (i) and (ii) made above are satisfied; in particular generates the nilpotent left-shift semigroup given by However, is not always a generator. For example if , then and ; hence is not a generator. Thus one needs an additional assumption on .

Definition 24. A bounded linear functional has little mass in if there exist and such that for every satisfying .

Note that and hence . Now the following holds.

Corollary 25. If has little mass in , then for all the operator is the generator of a strongly continuous semigroup on .

Proof. By Corollary 22 it suffices to show that for the triple the conditions (ii)–(v) of Theorem 10 are satisfied. To this end, note that and that the Dirichlet operator is given by where for all .
(ii) By Remark 2 it suffices to verify estimate (11) where we may assume that for some . Using integration by parts and [23, Theorem  4.2] we conclude that (for a function defined on an interval denote in the sequel by its extension to by the value ) This implies for all which shows (ii).
(iii) By the Riesz-Markov representation theorem there exists a regular complex Borel measure on such that Using Fubini’s theorem and Hölder’s inequality one obtains for and where (which coincides with ). This proves (iii).
(iv) From (84) one obtains for and by similar arguments as in (iii) This shows (iv).
(v) Since by assumption has little mass in , it follows that for sufficiently small . Hence from estimate (87) and the denseness of in it follows that for sufficiently small. This implies as claimed.

Remarks. (i) Corollary 25 could be easily generalized (with essentially the same proof) to the first derivative on . One could even go further and prove a similar result on for a (possibly infinite dimensional) Banach space provided the boundary operator has a representation as a Riemann-Stieltjes integral as in (85). See also [22, Example  5.1].

(ii) In most cases the admissibility of the identity as a feedback operator is verified by showing that for sufficiently small . Choosing , by (84), one obtains for all ; hence if and only if . This provides an example where our perturbation theorem is applicable even if for all . Note that for one obtains ; hence in this case cannot be a generator.

Appendix

Estimating the -Norm of a Triangular Toeplitz Matrix

For the proof of Lemma 12 one needed the following result.

Lemma A.1. For a Banach space endow , , with the -norm for some . Moreover, let . Then the norm of the Toeplitz operator matrix can be estimated as

Proof. Let . Then one can estimate where the second last step follows from Young’s inequality applied to the convolution of sequences.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Tuebingen University.