© Hindawi Publishing Corp. SPECTRAL INCLUSIONS AND STABILITY RESULTS FOR STRONGLY CONTINUOUS SEMIGROUPS

We prove some spectral inclusions for strongly continuous semigroups. Some stability results are also established.

1. Introduction and preliminaries.Let X be a complex Banach space and let A be a closed linear operator with domain D(A), kernel N(A), and range R(A) in X.We will say that A is semiregular if R(A) is closed and Denote the regular spectrum by σ γ (A) := {λ ∈ C, λ− A is not semiregular}. (1.1) The set σ γ (A) was studied (under various names) by several authors, see for instance [11,12,13,14] and the references therein.An operator A is said to be essentially semiregular if R(A) is closed and there exists a finite-dimensional subspace G ⊆ X such that N(A) ⊆ R ∞ (A) + G. Define further the essential regular spectrum of A by σ γe (A) := {λ ∈ C, λ− A is not essentially semiregular}. (1.2) This concept was introduced and studied for bounded operators in [13,17].We say that A is upper semi-Fredholm if R(A) is closed and dim N(A) < ∞.The left essential spectrum is given by σ π (A) := {λ ∈ C, λ− A is not upper semi-Fredholm}. (1.3) Let X denote the dual space of X and A the adjoint operator of A. Define the reduced minimum modulus γ(A) by setting γ(A) := inf Au d u, N (A) , u ∈ D(A) \ N(A) .(1.4) It is well known (see [9]) that γ(A) = γ(A ) and γ(A) > 0 if and only if R(A) is closed.Let H 0 (A) denote the quasinilpotent part of A given by Let -= (T (t)) t≥0 be a strongly continuous semigroup with generator A on X.
We will denote the type (growth bound) ofby ω 0 : Following [6], the semigroupis called bounded if there exists M ≥ 1 such that T (t) ≤ M for all t ≥ 0. Basic materials on semigroups may be found in [4,6,15].
In [5], we have studied the regular spectrum for strongly continuous semigroups.As a continuation of [5], the present paper deals with the essentially regular spectrum.Moreover, we establish some stability results for strongly continuous semigroups.
The present paper is organized as follows.In Section 2, we first prove that the spectral inclusion for semigroups remains true for the regular spectrum, the left essential spectrum, and the essentially regular spectrum (Theorem 2.1).Secondly, we give necessary and sufficient conditions for the generator of a strongly continuous semigroup to be semiregular (Theorem 2.3) and essentially semiregular (Theorem 2.5).
In Section 3, we derive some stability results for strongly continuous semigroups.Among other results, we give necessary and sufficient conditions for the generator of a bounded strongly continuous semigroup to have no pure imaginary point in its spectrum (Theorem 3.3).This, in particular, provides us with a spectral characterization of the strong stability of the ultrapower extension of a given semigroup.Finally, we discuss the strong stability of a strongly continuous semigroup via the behavior of the resolvent of its generator, on the imaginary axis.
Throughout this paper, we let σ (A), ρ(A), σ p (A), σ ap (A), and σ su (A) denote, respectively, the spectrum, the resolvent set, the point spectrum, the approximative spectrum, and the surjective spectrum of an operator A. For λ ∈ ρ(A), R(λ, A) denotes the resolvent operator (λ − A) −1 ∈ Ꮾ(X) of A, where Ꮾ(X) stands for the algebra of bounded linear operators on X.
For later use, we introduce the following operator acting on X and depending on the parameters λ ∈ C and t ≥ 0: (1.7) It is well known (see [15]) that I(λ, t) is a bounded linear operator on X and we have We conclude this section by the following result which we need in the sequel.

Spectral inclusions.
In this section, we study the regular spectrum and the essentially regular spectrum of the generator of a strongly continuous semigroup.We begin with the following spectral inclusions.
To prove this result, we need the following lemma.
Lemma 2.2.Let A be the generator of a strongly continuous semigroup (T (t)) t≥0 .Then, for all λ ∈ C, t ≥ 0, and n ∈ N, (i) Proof of Lemma 2.2.As mentioned before, I(λ, t) is a bounded linear operator on X and we have (2.3) Proceeding by induction, we get the desired result.The assertions (ii), (iii), and (iv) follow easily from (i).

Proof of Theorem 2.1
The regular spectrum.See [5].
The left essential spectrum.Let t 0 > 0 be fixed and suppose that e λt 0 ∈ σ π (T (t 0 )) for some λ ∈ C. We show that λ ∈ σ π (A).Using Lemma 2.2(iii), together with dim N(e λt 0 −T (t 0 )) < ∞, we infer that N(λ−A) is finite dimensional.Now, we prove that R(λ−A) is closed.Since N(e λt 0 −T (t 0 )) is finite dimensional, there exists a closed subspace Y of X such that N(e λt 0 −T (t 0 ))⊕Y = X.But, (λ−A)(N(e λt 0 −T (t 0 ))∩D(A)) is finite dimensional and therefore closed.Then, we need only to show that (λ−A)(Y ∩D(A)) is closed.From the closedgraph theorem and the closedness of R(e λt 0 − T (t 0 )), it follows that there is a constant C > 0 such that (2.4) From Lemma 2.2(i), we obtain that, for every x ∈ D(A), for some positive constant M. The combination of inequalities (2.4) and (2.5) gives us From the fact that λ − A is closed, the result follows.
The essential regular spectrum.Let t 0 > 0 be fixed and suppose that e λt 0 − T (t 0 ) is essentially semiregular for some λ ∈ C \{0}.We show that λ − A is essentially semiregular.To this end, consider the closed (T (t)) t≥0 -invariant subspace M := R ∞ (e λt 0 − T (t 0 )) of X and the quotient semigroup ( T (t)) t≥0 defined on X/M by with generator A defined by From Lemma 1.1, it follows that the operator e λt 0 − T (t 0 ) is upper semi-Fredholm.Thus, e λt 0 ∉ σ π ( T (t 0 )).By virtue of the precedent case, we get λ ∉ σ π ( A).In consequence, the operator λ − A is upper semi-Fredholm.Next, let π : X → X/M be the canonical projection.Using Lemma 2.2(ii), together with dim(N(λ − A)) < ∞, it can be verified that for a finite-dimensional subspace G of X.Now, we show that R(λ−A) is closed.
To do this, consider a sequence (u n ) n of elements of R(λ − A), which converges to u.Then, there exists a sequence (v n ) n of elements of D(A) such that (2.10) Accordingly, u ∈ R(λ − A).Consequently, the operator λ − A is essentially semiregular.This proves the theorem.
The next theorem gives, under suitable assumptions, necessary and sufficient conditions for the generator of a strongly continuous semigroup to be semiregular.The proof can be found in [5].

Theorem 2.3. Let (T (t)) t≥0 be a strongly continuous semigroup with generator A and type ω 0 . If (T (t)) t≥0 satisfies any of the following conditions:
(a) lim t→∞ (1/t) T (t) = 0; (b) |ω 0 | < γ(A), then the following assertions are equivalent: (i) A is semiregular; The following example shows that conditions (a) and (b) in Theorem 2.3 are needed for the conclusion.We conclude this section by the following result.Theorem 2.5.Let A be the generator of a strongly continuous semigroup (T (t)) t≥0 satisfying lim t→∞ (1/t) T (t) = 0.The following assertions are equivalent: (i) A is essentially semiregular; (ii) A is upper semi-Fredholm.

Proof. (i)⇒(ii).
Since A is essentially semiregular, there exists a finitedimensional subspace G of X such that N(A) ⊆ R ∞ (A) + G.As noticed in [13], we may assume that G ⊆ N(A).Let y ∈ N(A) and let x ∈ D(A) and g ∈ G such that y = Ax + g.Using Lemma 2.2(i), we infer that (2.11) Since lim t→∞ (1/t) T (t) = 0, then y = g.In consequence, N(A) = G.This is the desired result.(ii)⇒(i).Obvious.
The semigroupis, by construction, strongly continuous.Its generator A is given by The spectra of A and A are related as follows: Theorem 3.3.Let A be the generator of a bounded strongly continuous semigroup -= (T (t)) t≥0 .Then the following assertions are equivalent: Proof.(i)⇒(ii).It suffices to apply Theorem 2.3 to the rescaled semigroup (e −iλt T (t)) t≥0 whose generator is A − iλ. (ii)⇒(i).Obvious.
Remark 3.4.(1) It was shown in [3], under the hypothesis of Theorem 3.3, that the condition (2) In the general case, the condition σ (A) ∩ iR = ∅ does not characterize, even in Hilbert spaces, the strong stability of the semigroup generated by A. The translation semigroup T (t)f (x) := f (x + t), t ≥ 0, on L 2 (R + ) shows that this condition is not necessary for strong stability.In fact, this semigroup has the generator A = d/dx, and the spectrum of A is the left half plane {λ ∈ C : Reλ ≤ 0}, see [1, A.III, 2.4, page 66].Hence, σ (A) ∩ iR = iR but lim t→∞ T (t)f = 0 for every f ∈ L 2 (R + ).However, Theorem 3.3 shows that the condition σ (A) ∩ iR = ∅ characterizes completely the strong stability of the ultrapower extension of the semigroup generated by A. Corollary 3.5.Let A be the generator of a bounded strongly continuous semigroup -= (T (t)) t≥0 on a reflexive Banach space X.Then, conditions (i), (ii), (iii), (iv), and (v) of Theorem 3.3 are equivalent to (vi) for every x ∈ X and for every β ∈ R, R(λ + iβ, A)x = O(1) (as λ → 0).
Proof.It is well known [19,Corollary 1.3.2]that the adjoint semigroup of a strongly continuous semigroup on a reflexive Banach space is again strongly continuous.It suffices to apply Theorem 3.3 to the adjoint semigroup whose generator is A .
Note that, in reflexive Banach spaces, condition (vi) implies the strong stability of the bounded semigroup generated by A (this follows from Corollary 3.5 and Theorem 3.1).In general Banach spaces setting, we have the following proposition.

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