Spectral Inclusions Between C0-Quasi-Semigroups and Their Generators

In this paper, we show a spectral inclusion of a different spectra of a C0-quasi-semigroup and its generator and precisely for ordinary, point, approximate point, residual, essential and regular spectra.


Introduction and preliminaries
Let X be a complex Banach space and B(X) the algebra of all bounded linear operators on X. We denote by D(T ), Rg(T ), Rg ∞ (T ) := ∩ n≥1 Rg(T n ), N (T ), ρ(T ), σ(T ), and σ p (T ) respectively the domain, the range, the hyper range, the kernel, the resolvent and the spectrum of T , where σ(T ) = {λ ∈ C \ λ−T is not bijective}.
The point , the approximate point , the residual and regular spectrum spectra are defined by • σ p (T ) = {λ ∈ C \ λ − T is not injective } • σ a (T ) = {λ ∈ C \ λ − T is not injective or Rg(λ − T ) is not closed in X }. and from [7, lemma 1.9] λ is an approximate eigenvalue,if and only if there exists a sequence (x n ) n∈N ⊂ D(A) called an approximate eigenvector , such that x n = 1 and lim n→∞ Ax n − λx n = 0.
An operator T ∈ B(X) is called Fredholm operator, in symbol T ∈ Φ(X), if α(T ) = dimN (T ) and β(T ) = codimRg(T ) are finite, and the essential spectrum is defined by, The family (T (t)) t≥0 ⊆ B(X) is a C 0 -semigroup if it has the following properties: In this case, its generator A is defined by The theory of quasi-semigroups of bounded linear operators, as a generalization of semigroups of operators, was introduced by Leiva and Barcenas [3] , [4] , [5]. (1) R(t, 0) = I, the identity operator on X, (2) R(t, s + r) = R(t + r, s)R(t, r), s≥0 on a Banach space X, let D be the set of all x ∈ X for which the following limits exist, The family {A(t)} t≥0 is called infinitesimal generator of the C 0 -quasi-semigroups {R(t, s)} t,s≥0 .
Throughout this paper we denote T (t) and R(t, s) as C 0 -semigroups {T (t)} t≥0 and C 0 -quasi-semigroup {R(t, s)} t,s≥0 respectively.We also denote D as domain for A(t) , t ≥ 0. .
(4) Since T (t) is strongly continuous on X , there exists ω and M ω > 0 such that , Moreover, The following results are obtained recently by Sutrima , Ch. Rini Indrati and others [12] , there are show some relations between a C 0 -quasi-semigroup and its generator.
(2) For each t ≥ 0 and x ∈ X, is locally integrable, then for every x ∈ D and s ≥ 0, In this work, we show that the spectral inclusion of different spectra of C 0 -semigroups valid for C 0 -quasisemigroups.

Main results
For later use, we introduce the following operator acting on X and depending on the parameters λ ∈ C and t, s ≥ 0 : (2) For all x ∈ D,

Proof.
(1) For all x ∈ X we have And we obtain , Then lim r→0 + R(0,r)D λ (t,s)x−D λ (t,s)x r exists.
And, Hence, we deduce that D λ (t, s)x ∈ D And , (2) For all x ∈ D and all t, s ≥ 0 we have , Corollary 2.1. In the case of C 0 -semigroup T (s) = R(t, s) , we retrieve the equality [9], . Then for all λ ∈ C , t, s ≥ 0 and n ∈ N , Proof. follow easily from , The following theorem characterizes the ordinary, point, approximate point, essential and residual spectra of a C 0 -quasi-semigroup.
By Theorem 2.1, we obtain for every x ∈ D On the other hand, also from Theorem 2.1, we obtain for every x ∈ X x = [e λs − R(t, s)]F λ (t, s)x; Since we know that R(t, s)F λ (t, s) = F λ (t, s)R(t, s), then Finally, we conclude that λ − A(t) is invertible and hence λ / ∈ σ(A(t)).
(3) Let λ ∈ σ a (A(t)) and a corresponding approximate eigenvector (x n ) n∈N ⊂ D , we define the sequence (y n ) by y n := e λs x n − R(t, s)x n By theorem 2.1 we have So there is a constant c 0 such that , Hence, e λs is an approximate eigenvalue of R(t, s), and (x n ) n∈N serves as the same approximate eigenvector for all t, s ≥ 0 (4) Let λ ∈ C such that e λt / ∈ σ e (R(t, s))).
is not dense , now we use the corollary 2.2 we obtain Thus Rg[e λs − R(t, s)] is not dense and finally e λs ∈ σ r (R(t, s)).
In the next theorem, we will prove that the spectral inclusion of C 0 -quasi-semigroups remains true for the regular spectrum. To prove this result, we need the following proposition and lemma.
induced by T is bounded below.
In consequence, the operator λ − A(t) is injective and the closed range and from corollary 2.2 (7) Then , Now, we show that Rg(λ − A(t))is closed.
To do this, consider a sequence (u n ) n≥0 of elements of Rg(λ − A(t)) , which converges to u. Then, there exists a sequence (v n ) n≥0 of elements of D such that (λ − A(t))u n = v n → u. Since Rg(λ − A(t)) is closed, there exists w ∈ D such that u = (λ − A(t)) w.
In the next work we will try to demonstrate the equality of the spectra or give counter-examples in the case of a strict inclusion.