$C_0$ Semigroup and Local Spectral Theory

In this paper, we studied some local spectral properties for a Co semigroup and its generator. Some stabilities results are also established.


Introduction
The semigroups can be used to solve a large class of problems commonly known as the Cauchy problem, u ′ (t) = Au(t) for all t ≥ 0, u(0) = u 0 . . on a Banach space X. Here A is a given linear operator with domain D(A) and the initial value u 0 . The solution of the previous Cauchy problem will be given by u(t) = T (t)u 0 for an operator semigroup (T (t)) t≥0 on X. In this paper, we will focus on a special class of linear semigroups called C 0 semigroups which are semigroups of strongly continuous bounded operators. Precisely, a one-parameter family (T (t)) t≥0 of operators on a Banach space X is called a C 0 -semigroup of operators or a strongly continuous semigroup of operators if, 1. T (0) = I, 2. T (t + s) = T (t)T (s), ∀t, s ≥ 0, (T (t)) t≥0 has a unique infinitesimal generator A defined in domain D(A) by, Also, T (t) are linear and continuous operators on X for all t ≥ 0, and A is a closed operator, see [4,8]. In order to understand the behavior of the solutions in terms of the data concerning A, one seeks information about the spectrum of T (t) in terms of the spectrum of A. Unfortunately the spectral mapping theorem e tσ(A) = σ(T (t)) \ {0} often fails, sometimes in dramatic ways. However, the inclusion is always true. The aim of this paper is to develop a local spectral theory for C 0 semigroups.

Preliminaries
Throughout, X denotes a complex Banach space, let A be a closed operator on X with domain D(A). We denote by A * , R(A), N (A), R ∞ (A) = n≥0 R(A n ), σ K (A), σ su (A), σ(A), respectively the adjoint, the range, the null space, the hyper-range, the semi-regular spectrum, the surjectivity spectrum and the spectrum of A. Recall that for a closed operator A and x ∈ X, the local resolvent of A at x, ρ A (x) defined as the union of all open subset U of C for which there is an analytic function f : [5,7]. For any arbitrary closed set Ω in the complex field, the spectral subspace associated to Ω is : is a hyperinvariant subspace of A not always closed, see [6].
Next, let A be a closed operator, A is said to have the single valued extension property at λ 0 ∈ C (SVEP) if for every open disc D λ 0 ⊆ C centered at λ 0 , the only analytic function f : [5]. Let (T (t)) t≥0 be a C 0 semigroup with generator A, we introduce the following operator acting on X and depending on the parameters λ ∈ C and t ≥ 0, It is well known that B λ (t) is a bounded operator on X and we have ( [4,8]): Recall that some spectral inclusions for various reduced spectra are studied in [3], [4] and [8]. The authors proved that where ν ∈ {σ ap , σ K }, approximate point spectrum and semi-regular spectrum, also we have equality where ν ∈ {σ p , σ r } point spectrum and residual spectrum. In the next two sections, we will prove a spectral inclusion for local spectrum and a framing of S(; ) which characterizes it. Some related stability results are also presented.
In the following, we give a sufficient condition to show that the spectral subspace X A (∅) is closed for all t > 0. 1. If T (t 0 ) has the SVEP for some t 0 ≥ 0, then A has the SVEP.

Example 3.5.
A C 0 -semigroup (T (t)) t≥0 is called periodic if there exists t 0 > 0 such that T (t 0 ) = I, so T (t 0 ) has the SVEP. From corollary 3.3, the infinitesimal generator A of (T (t)) t≥0 has the SVEP.
To continue the development of a spectral theory for semigroups and theirs generators, we prove that the formula (1.1) holds for local spectrum. Theorem 3.6. Let (T (t)) t≥0 be a C 0 -semigroup on X with infinitesimal generator A. The following spectral inclusion holds : H. Boua, M. Karmouni and A. Tajmouati and sup y i Remark 3.1. The spectral inclusion for local spectrum is strict. Indeed, let (T (t)) t≥0 be a quasi-nilpotent C 0 semigroup with infinitesimal generator A, and 0 = x ∈ X. We have σ T (t) (x) = {0}, but e tσA(x) = ∅.
In [2], A. Elkoutri and M. A. Taoudi showed that (T (t)) t≥0 is strongly stable if σ K (A) ∩ iR = ∅. In the following, we give a stability result for strongly continuous semigroups using the local spectrum: