Semi-Fredholm and Semi-Browder Spectra For $C_0$-Quasi-Semigroups

TL;DR

This paper investigates various spectral properties of $C_0$-quasi-semigroups, linking their spectra to those of their generators, with a focus on essential ascent, descent, and semi-Fredholm and semi-Browder spectra.

Contribution

It provides a detailed description of the spectra of $C_0$-quasi-semigroups in relation to their generators, including semi-Fredholm and semi-Browder spectra, which was not previously fully characterized.

Findings

Spectral descriptions of $C_0$-quasi-semigroups in terms of their generators.

Characterization of essential ascent and descent spectra.

Analysis of Drazin invertibility and semi-Fredholm spectra.

Abstract

In this paper, we describe the different spectra of the $C_0$-quasi-semigroups by the spectra of their generators. Specially, essential ascent and descent,Drazin invertible, upper and lower semi-Fredholm and semi-Browder spectra.

Equations

$${\sigma }_D(T)=\{\lambda \in \mathbb{C}/\lambda -T\text{mbox}sinotDrazininvertible\}$$

$$(\lambda -A(t))D_{\lambda }(t,s)x=[e^{\lambda s}-R(t,s)]x.$$

$$D_{\lambda }(t,s)(\lambda -A(t))x=[e^{\lambda s}-R(t,s)]x.$$

$$R(0,r)D_{\lambda }(t,s)x$$

$$\underset{r\to 0^{+}}{\lim }\frac{R(0,r)D_{\lambda }(t,s)x-D_{\lambda }(t,s)x}{r}$$

$$\underset{r\to 0^{+}}{\lim }\frac{R(t,r)D_{\lambda }(t,s)x-D_{\lambda }(t,s)x}{r}$$

$$\underset{r\to 0^{+}}{\lim }\frac{R(t-r,r)D_{\lambda }(t,s)x-D_{\lambda }(t,s)x}{r}$$

$$A(t)D_{\lambda }(t,s)x$$

$$D_{\lambda }(t,s)A(t)x$$

$$e^{\lambda s}x-R(t,s)x$$

$$(\lambda -A(t){)}^n{K}_{\lambda ,n}(t,s)+[D_{\lambda ,n}(t,s){]}^n[D_{\lambda }(t,s){]}^n=s^{n^2}.$$

$$L_{\lambda }(t,s)x$$

$$(\mu -A(t))L_{\lambda }(t,s)x=\mu L_{\lambda }(t,s)x-{\int }_0^s{e}^{-\lambda h}A(t)D_{\lambda }(t,h)xdh.$$

$$A(t)L_{\lambda }(t,s)x={\int }_0^s{e}^{-\lambda h}A(t)D_{\lambda }(t,h)xdh.$$

$$(\lambda -A(t))L_{\lambda }(t,s)x$$

$$D_{\lambda }(t,s)D_{\lambda }(t,r)$$

$$D_{\lambda }(t,s)L_{\lambda }(t,s)$$

$$(\lambda -A(t))L_{\lambda }(t,s)x$$

$$[(\lambda -A(t))L_{\lambda }(t,s){]}^n$$

$$D_{\lambda ,n}(t,s)=\sum _{i=1}^n{C}_n^i{s}^{n-i}[{\phi }_{\lambda }(s){]}^i[-D_{\lambda }(t,s){]}^{i-1}.$$

$$(\lambda -A(t){)}^n[L_{\lambda }(t,s){]}^n+D_{\lambda }(t,s)D_{\lambda ,n}(t,s)=s^n.$$

$$[D_{\lambda }(t,s)D_{\lambda ,n}(t,s){]}^n$$

$$[D_{\lambda }(t,s){]}^n[D_{\lambda ,n}(t,s){]}^n+(\lambda -A(t){)}^n{K}_{\lambda ,n}(t,s)=s^{n^2}.$$

$$(\lambda -A(t){)}^n{x}_n=y_n.$$

$$(\lambda -A(t){)}^n{K}_{\lambda ,n}(t,s)+[D_{\lambda ,n}(t,s){]}^n[D_{\lambda }(t,s){]}^n=s^{n^2}.$$

$$[e^{\lambda s}-R(t,s){]}^n[D_{\lambda ,n}(t,s){]}^n{x}_n$$

$$s^{n^2}{y}_n-(\lambda -A(t){)}^n{K}_{\lambda ,n}(t,s)y_n\in Rg[e^{\lambda s}-R(t,s)].$$

$$s^{n^2}y-(\lambda -A(t){)}^n{K}_{\lambda ,n}(t,s)y\in Rg[e^{\lambda s}-R(t,s)].$$

$$[e^{\lambda s}-R(t,s){]}^{n}z=s^{n^2}y-(\lambda -A(t){)}^n{K}_{\lambda ,n}(t,s)y.$$

$$y$$

Full text

Semi-Fredholm and Semi-Browder Spectra For $C_{0}$-quasi-semigroups

A. Tajmouati   ,   Y. Zahouan

A. Tajmouati, Y. Zahouan

Sidi Mohamed Ben Abdellah Univeristy, Faculty of Sciences Dhar Al Mahraz, Fez, Morocco.

abdelaziz.tajmouati@usmba.ac.ma

zahouanyouness1@gmail.com

Abstract

In this paper, we describe the different spectra of the $C_{0}$-quasi-semigroups by the spectra of their generators. Specially, essential ascent and descent,Drazin invertible, upper and lower semi-Fredholm and semi-Browder spectra.

keywords: $C_{0}$-quasi-semigroup,$C_{0}$-semigroup ,semi-Fredholm , ascent , descent spectrum , Drazin spectrum , ,semi-Browder spectrum.

1. Introduction

Let $X$ be a complex Banach space and $\mathcal{B}(X)$ the algebra of all bounded linear operators on $X$. We denote by $Rg(T)$, $N(T)$, $\rho(T)$ and $\sigma(T),$ respectively the range, the kernel, the resolvent and the spectrum of $T$, where $\sigma(T)=\{\lambda\in\mathbb{C}\,/\,\lambda-T\,\mbox{is not bijective}\}$. The function resolvent of $T\in\mathcal{B}(X)$ is defined for all $\lambda\in\rho(T)$ by $\mathcal{R}(\lambda,T)=(\lambda-T)^{-1}.$ The ascent and descent of an operator $T$ are defined respectively by,

$a(T)=\min\{k\in\mathbb{N}\,/\,N(T^{k})=N(T^{k+1})\}\;;\;d(T)=\min\{k\in\mathbb{N}\,/\,Rg(T^{k})=Rg(T^{k+1})\}$.

with the convention $inf(\varnothing)=\infty$.

The essential ascent and descent of an operator $T$ are defined respectively by,

$a_{e}(T)=\min\{k\in\mathbb{N}\,/\,dim[N(T^{k+1})/N(T^{k})]<\infty\}$ ; $d_{e}(T)=\min\{k\in\mathbb{N}/\,dim[Rg(T^{k})/Rg(T^{k+1})]<\infty\}$.

The ascent, descent , essential ascent and essential descent spectra are defined by,

• •

$\sigma_{a}(T)=\{\lambda\in\mathbb{C}\,/\,a(\lambda-T)=\infty\}$     ;   $\sigma_{d}(T)=\{\lambda\in\mathbb{C}\,/\,d(\lambda-T)=\infty\}$.

• •

$\sigma_{a_{e}}(T)=\{\lambda\in\mathbb{C}\,/\,a_{e}(\lambda-T)=\infty\}$   ;   $\sigma_{d_{e}}(T)=\{\lambda\in\mathbb{C}\,/\,d_{e}(\lambda-T)=\infty\}$.

We say that a closed linear operator $A$ is Drazin invertible if $d(T)=a(T)=p<\infty$ and $Rg(A^{p})$ is closed. The Drazin spectrum is

[TABLE]

The sets of upper and lower semi-Fredholm and their spectra are defined respectively by,

• •

$\Phi_{+}(X)=\{T\in\mathcal{B}(X)\,/\,\alpha(T):=dim(N(T))<\infty\;\mbox{and}\;Rg(T)\,\mbox{is closed}\}$

$\sigma_{e_{+}}(T)=\{\lambda\in\mathbb{C}\,/\,\lambda-T\notin\Phi_{+}(X)\}$

• •

$\Phi_{-}(X)=\{T\in\mathcal{B}(X)\,/\,\beta(T):=codim(Rg(T))=dim(X\backslash Rg(T))<+\infty\}$

$\sigma_{e_{-}}(T)=\{\lambda\in\mathbb{C}/\lambda-T\notin\Phi_{-}(X)\}.$

An operator $T\in\mathcal{B}(X)$ is called semi-Fredholm, in symbol $T\in\Phi_{\pm}(X)$, if $T\in\Phi_{+}(X)\cup\Phi_{-}(X).$

An operator $T\in\mathcal{B}(X)$ is called Fredholm, in symbol $T\in\Phi(X)$, if $T\in\Phi_{+}(X)\cap\Phi_{-}(X)$.

The essential and semi-Fredholm spectra are defined by,

• •

$\sigma_{e}(T)=\{\lambda\in\mathbb{C}/\lambda-T\notin\Phi(X)\}$

• •

$\sigma_{e_{\pm}}(T)=\{\lambda\in\mathbb{C}/\lambda-T\notin\Phi_{\pm}(X)\}$

The sets of upper and lower semi-Browder and their spectra are defined respectively by,

• •

${Br}_{+}(X)=\{T\in\Phi_{+}(X)\,/\,a(T)<+\infty\}$ ; $\sigma_{{Br}_{+}}(T)=\{\lambda\in\mathbb{C}/\lambda-T\notin Br_{+}(X)\}$

• •

${Br}_{-}(X)=\{T\in\Phi_{-}(X)\,/\,d(T)<+\infty\}$ ; $\sigma_{Br_{-}}(T)=\{\lambda\in\mathbb{C}/\lambda-T\notin Br_{-}(X)\}$

An operator $T\in\mathcal{B}(X)$ is called semi-Browder, in symbol $T\in Br_{\pm}(X)$, if $T\in{Br}_{+}(X)\cup{Br}_{-}(X)$.

An operator $T\in\mathcal{B}(X)$ is called Browder, in symbol $T\in Br(X)$, if $T\in Br_{+}(X)\cap Br_{-}(X)$.

The semi-Browder and Browder spectra are defined by,

• •

$\sigma_{{Br}_{\pm}}(T)=\{\lambda\in\mathbb{C}/\lambda-T\notin{Br}_{\pm}(X)\}$.

• •

$\sigma_{Br}(T)=\{\lambda\in\mathbb{C}/\lambda-T\notin Br(X)\}$.

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].Recently Sutrima , Ch. Rini Indrati and others [9] , there are show some relations between a $C_{0}$-quasi-semigroup and its generator related to the time-dependent evolution equation.

Definition 1.1**.**

[3]** Let $X$ be a complex Banach. The family $\left\{R(t,s)\right\}_{t,s\geq 0}\subseteq\mathcal{B}(X)$ is called a strongly continuous quasi-semigroup (or $C_{0}$-quasi-semigroup) of operators if for every $t,s,r\geq 0$ and $x\in X$ ,

1. (1)

$R(t,0)=I$, the identity operator on $X$, 2. (2)

$R(t,s+r)=R(t+r,s)R(t,r)$, 3. (3)

$lim_{s\longrightarrow 0}\left|\left|R(t,s)x-x\right|\right|=0$* ,* 4. (4)

There exists a continuous increasing mapping $M:[0;+\infty[\longrightarrow[1;+\infty[$ such that ,

$\left|\left|R(t,s)\right|\right|\leq M(t+s)$**

Definition 1.2**.**

[3]** For a $C_{0}$-quasi-semigroup $\left\{R(t,s)\right\}_{t,s\geq 0}$ on a Banach space $X$, let $\mathcal{D}$ be the set of all $x\in X$ for which the following limits exist,

$\lim_{s\rightarrow 0^{+}}\frac{R(0,s)x-x}{s}$*     and     $\lim_{s\rightarrow 0^{+}}\frac{R(t,s)x-x}{s}=\lim_{s\rightarrow 0^{+}}\frac{R(t-s,s)x-x}{s}$ ,   $t>0$*

• •

For $t\geq 0$ we define an operator $A(t)$ on $\mathcal{D}$ as $A(t)x=\lim_{s\rightarrow 0^{+}}\frac{R(t,s)x-x}{s}$

The family $\left\{A(t)\right\}_{t\geq 0}$ is called infinitesimal generator of the $C_{0}$-quasi-semigroups $\left\{R(t,s)\right\}_{t,s\geq 0}$.

• •

For each $t\geq 0$ we define the resolvent operator of $A(t)$ as $\mathcal{R}(\lambda,A(t))=(\lambda I-A(t))^{-1}$, with its resolvent set $\rho(A(t))$.

Throughout this paper we denote $T(t)$ and $R(t,s)$ as $C_{0}$-semigroups $\left\{T(t)\right\}_{t\geq 0}$ and $C_{0}$-quasi-semigroup $\left\{R(t,s)\right\}_{t,s\geq 0}$ respectively.We also denote $\mathcal{D}$ as domain for $A(t)$ , $t\geq 0$.

Theorem 1.1**.**

[9]**

Let $R(t,s)$ is a $C_{0}$-quasi-semigroup on $X$ with generator $A(t)$ then,

1. (1)

For each $t\geq 0$ , $R(t,.)$ is strongly continuous on $[0;+\infty[$. 2. (2)

For each $t\geq 0$ and $x\in X$,

$\lim_{s\rightarrow 0^{+}}\frac{1}{s}\int_{0}^{s}R(t,h)xdh=x$** 3. (3)

If $x\in\mathcal{D}$ , $t\geq 0$ and $t_{0},s_{0}\geq 0$ then, $R(t_{0},s_{0})x\in\mathcal{D}$ and

$R(t_{0},s_{0})A(t)x=A(t)R(t_{0},s_{0})x$** 4. (4)

For each $s>0$ , $\frac{\partial}{\partial s}(R(t,s)x)=A(t+s)R(t,s)x=R(t,s)A(t+s)x$; $x\in\mathcal{D}.$ 5. (5)

If $A(.)$ is locally integrable, then for every $x\in\mathcal{D}$ and $s\geq 0$,

$R(t,s)x=x+\int_{0}^{s}A(t+h)R(t,h)xdh.$** 6. (6)

If $f:[0;+\infty[\longrightarrow X$ is a continuous, then for every $t\in[0;+\infty[$

$\lim_{r\rightarrow 0^{+}}\int_{s}^{s+r}R(t,h)f(h)xdh=R(t,s)f(s)$**

In the semigroups theory, if $A$ is an infinitesimal generator of $C_{0}$-semigroup with domain $D(A)$ , then $A$ is a closed operator and $D(A)$ is dense in $X$. These are is not always true for any $C_{0}$-quasi-semigroups, see [9].

In [6],[7],[8],[10],[11], the authors have studied the different spectra of the $C_{0}$-semigroups. In our paper [13], we have studied ordinary , point , approximate point , residual , essential and regular spectra of the $C_{0}$-quasi-semigroups. In this paper, we continue to study $C_{0}$-quasi-semigroups. We investigate the relationships between the different spectra of the $C_{0}$-quasi-semigroups and their generators, precisely the essential ascent and descent, Drazin, upper and lower semi-Fredholm and semi-Browder spectra.

2. Main results

In [13] we have proved the following theorem,

Theorem 2.1**.**

Let $A(t)$ be the generator of the C${}_{0}-$quasi-semigroup $(R(t,s))_{t,s\geq 0}$ . Then for all $\lambda\in\mathbb{C}$ and all $t,s\geq 0$ we have

1. (1)

For all $x\in X$ we have

[TABLE] 2. (2)

For all $x\in\mathcal{D}$ we have

[TABLE]

With $D_{\lambda}(t,s)x=\int_{0}^{s}e^{\lambda(s-h)}R(t,h)xdh$ for all $x\in X$ and $t,s\geq 0$ is a bounded linear operator on $X$.

Proof.

1. (1)

For all $x\in X$ we have

[TABLE]

And we obtain ,

[TABLE]

Then $\lim_{r\rightarrow 0^{+}}\frac{R(0,r)D_{\lambda}(t,s)x-D_{\lambda}(t,s)x}{r}$ exists.

And,

[TABLE]

Moreover,

[TABLE]

Thus , $\lim_{r\rightarrow 0^{+}}\frac{R(t,r)D_{\lambda}(t,s)x-D_{\lambda}(t,s)x}{r}=\lim_{r\rightarrow 0^{+}}\frac{R(t-r,r)D_{\lambda}(t,s)x-D_{\lambda}(t,s)x}{r}$

Hence, we deduce that $D_{\lambda}(t,s)x\in\mathcal{D}$ And ,

[TABLE]

Finaly , $(\lambda-A(t))D_{\lambda}(t,s)x=[e^{\lambda s}-R(t,s)]x$ for all $x\in X$. 2. (2)

For all $x\in\mathcal{D}$ and all $t,s\geq 0$ we have ,

[TABLE]

Thus, we deduce for all $x\in D(A)$ $D_{\lambda}(t,s)(\lambda-A(t))x=[e^{\lambda s}-R(t,s)]x$.

∎

Corollary 2.1**.**

Let $A(t)$ be the generator of a $C_{0}$-quasi-semigroup $(R(t,s))_{t,s\geq 0}$ . Then for all $\lambda\in\mathbb{C}$ , $t,s\geq 0$ and $n\in\mathbb{N}$ ,

1. (1)

For all $x\in X$ ,

$(\lambda-A(t))^{n}[D_{\lambda}(t,s)]^{n}x=[e^{\lambda s}-R(t,s)]^{n}x$. 2. (2)

For all $x\in\mathcal{D}^{n}$   (Domain of $A(t)^{n}$) ,

$[D_{\lambda}(t,s)]^{n}(\lambda-A(t)^{n})x=[e^{\lambda s}-R(t,s)]^{n}x.$** 3. (3)

$N[\lambda-A(t)]\subseteq N[e^{\lambda s}-R(t,s)].$** 4. (4)

$Rg[e^{\lambda s}-R(t,s)]\subseteq Rg[\lambda-A(t)].$** 5. (5)

$N[\lambda-A(t)]^{n}\subseteq N[e^{\lambda s}-R(t,s)]^{n}.$** 6. (6)

$Rg[e^{\lambda s}-R(t,s)]^{n}\subseteq Rg[\lambda-A(t)]^{n}.$

Proof.

follow easily from ,

[TABLE]

∎

To obtain the results concerning the semi-Fredholm and semi-Browder spectra we need the following theorem.

Theorem 2.2**.**

Let $A(t)$ be a closed and densely defined generator of a $C_{0}-$quasi-semigroup $(R(t,s))_{t,s\geq 0}$ on a Banach space $X$ . Then for all $\lambda\in\mathbb{C}$ and all $t,s\geq 0$ we have,

1. (1)

$(\lambda-A(t))L_{\lambda}(t,s)+\varphi_{\lambda}(s)D_{\lambda}(t,s)=\phi_{\lambda}(s)I,$* where $L_{\lambda}(t,s)=\int_{0}^{s}e^{-\lambda h}D_{\lambda}(t,h)dh\,\,$ , $\,\,\varphi_{\lambda}(s)=e^{-\lambda s}$ and $\phi_{\lambda}(s)=s$.*

Moreover, the operators $L_{\lambda}(t,s),$ $D_{\lambda}(t,s)$ and $(\lambda-A(t))$ are mutually commuting. 2. (2)

For all $n\in\mathbb{N}^{*},$ there exists an operator $D_{\lambda,n}(t,s)\in\mathcal{B}(X)$ such that,

$(\lambda-A(t))^{n}[L_{\lambda}(t,s)]^{n}+D_{\lambda,n}(t,s)D_{\lambda}(t,s)=s^{n}.$**

Moreover, the operator $D_{\lambda,n}(t,s)$ is commute with each one of $D_{\lambda}(t,s)\,$ and $\,L_{\lambda}(t,s)$. 3. (3)

For all $n\in\mathbb{N}^{*},$ there exists an operator $K_{\lambda,n}(t,s)\in\mathcal{B}(X)$ such that,

[TABLE]

Moreover, the operator $K_{\lambda,n}(t,s)$ is commute with each one of $D_{\lambda}(t,s)\,$ and $\,D_{\lambda,n}(t,s)$.

Proof.

1. (1)

Let $\mu\in\rho(A(t))$. By theorem 2.2, for all $x\in X$ we have $D_{\lambda}(t,h)x\in\mathcal{D}$ and hence,for all $t,s\geq 0$,

[TABLE]

Therefore for all $x\in X$, we have $L_{\lambda}(t,s)x\in\mathcal{D}$ and

[TABLE]

Thus

[TABLE]

Hence, we conclude that

[TABLE]

where $\phi_{\lambda}(s)=s$ and $\varphi_{\lambda}(s)=e^{-\lambda s}.$

Therefore, we obtain $(\lambda-A(t))L_{\lambda}(t,s)+\varphi_{\lambda}(s)D_{\lambda}(t,s)=\phi_{\lambda}(s)I.$

On the other hand,for all $s>t<r\geq 0$ we have ,$R(t,s)R(t,r)=R(t-s,s)R(t,r)=R(t-s-r,s+r)$ and $R(t,r)R(t,s)=R(t-r,r)R(t,s)=R(t-r-s-r,s+r)$ Then $R(t,s)R(t,r)=R(t,r)R(t,s)$ for all $s,t,r\geq 0$, then $D_{\lambda}(t,h)R(t,s)=R(t,s)D_{\lambda}(t,h).$

Hence

[TABLE]

Thus, we deduce that

[TABLE]

Since for all $x\in X$, $A(t)L_{\lambda}(t,s)x=\int_{0}^{s}e^{-\lambda h}A(t)D_{\lambda}(t,h)xdh$ and for all $x\in\mathcal{D}$, $A(t)D_{\lambda}(t,h)x=D_{\lambda}(t,h)A(t)x,$ then we obtain for all $x\in\mathcal{D}$,

[TABLE] 2. (2)

For all $n\in\mathbb{N}^{*}$, we obtain

[TABLE]

where

[TABLE]

Therefore, we have

[TABLE]

Finally, for commutativity, it is clear that $D_{\lambda,n}(t,s)$ commute with each one of $D_{\lambda}(t,s)\,$ and $\,L_{\lambda}(t,s)$ since the operators $L_{\lambda}(t,s),$ $D_{\lambda}(t,s)$ and $(\lambda-A(t))$ are mutually commuting from (1). 3. (3)

Since we have $D_{\lambda}(t,s)D_{\lambda,n}(t,s)=s^{n}-(\lambda-A(t))^{n}[L_{\lambda}(t,s)]^{n},$ then for all $n\in\mathbb{N}$

[TABLE]

where $K_{\lambda,n}(t,s)=\sum_{i=1}^{n}C_{n}^{i}s^{n(n-i)}(\lambda-A(t))^{n(i-1)}[L_{\lambda}(t,s)]^{ni}.$ Hence, we obtain

[TABLE]

Finally, the commutativity is clear.

∎

We start by this result.

Proposition 2.1**.**

Let $A(t)$ be a closed and densely defined generator of a $C_{0}-$quasi-semigroup $(R(t,s))_{t,s\geq 0}$ on a Banach space $X$. If   $Rg[e^{\lambda s}-R(t,s)]^{n}$ is closed, then $Rg[\lambda-A(t)]^{n}$ is also closed.

Proof.

Let $(y_{n})_{n\in\mathbb{N}}\subseteq X$ such that $y_{n}\rightarrow y\in X$ and there exists $(x_{n})_{n\in\mathbb{N}}\subseteq\mathcal{D}$ satisfying

[TABLE]

By theorem 2.2, for all $n\in\mathbb{N}^{*},$ there exists $D_{\lambda,n}(t,s)$, $K_{\lambda,n}(t,s)\in\mathcal{B}(X)$ such that,

[TABLE]

Hence, we conclude that

[TABLE]

Thus,

[TABLE]

Therefore, since $Rg[e^{\lambda s}-R(t,s)]^{n}$ is closed, $K_{\lambda,n}(t,s)$ is bounded linear and $s^{n^{2}}y_{n}-(\lambda-A(t))^{n}K_{\lambda,n}(t,s)y_{n}$ converges to $s^{n^{2}}y-(\lambda-A(t))^{n}K_{\lambda,n}(t,s)y,$ we conclude that

[TABLE]

Then there exists $z\in X$ such that

[TABLE]

Hence for all $s\neq 0$, we have

[TABLE]

Finally, we obtain

[TABLE]

∎

The following result discusses the semi-Fredholm spectrum.

Theorem 2.3**.**

Let $A(t)$ be a closed and densely defined generator of a $C_{0}-$quasi-semigroup $(R(t,s))_{t,s\geq 0}$ on a Banach space $X$. For all $\lambda\in\mathbb{C}$ and all $t,s\geq 0$, we have

1. (1)

$e^{\sigma_{e_{+}}(A(t))s}\subseteq\sigma_{e_{+}}(R(t,s));$** 2. (2)

$e^{\sigma_{e_{-}}(A(t))s}\subseteq\sigma_{e_{-}}(R(t,s));$** 3. (3)

$e^{\sigma_{e_{\pm}(A(t))s}}\subseteq\sigma_{e_{\pm}}(R(t,s)).$**

Proof.

1. (1)

Suppose that $e^{\lambda s}\notin\sigma_{e_{+}}(R(t,s))$, then there exists $n\in\mathbb{N}$ such that $\alpha[e^{\lambda s}-R(t,s)]=n$ and $Rg[e^{\lambda s}-R(t,s)]$ is closed.

By theorem 2.1, we obtain

[TABLE]

then

[TABLE]

On the other hand, from Proposition 2.1, we deduce that $Rg(\lambda-A(t))$ is closed.

Therefore $\lambda-A(t)\in\Phi_{+}(\mathcal{D})$, Then$\lambda\notin\sigma_{e_{+}}(A(t)).$ 2. (2)

Suppose that $e^{\lambda s}\notin\sigma_{e_{-}}(R(t,s))$, then there exist $n\in\mathbb{N}$ such that $\beta[e^{\lambda s}-R(t,s)]=n$.

By theorem 2.1, we obtain

[TABLE]

then $\beta(\lambda-A(t))\leq n$ and hence, $\lambda\notin\sigma_{e_{-}}(A(t))$ 3. (3)

It is automatic by the previous assertions of this theorem.

∎

Proposition 2.2**.**

Let $A(t)$ be the generator of a $C_{0}$-quasi-semigroup $(R(t,s))_{t,s\geq 0}$. For all $\lambda\in\mathbb{C}$ and all $t,s\geq 0$,we have

1. (1)

If $d[e^{\lambda s}-R(t,s)]=n,$ then $d[\lambda-A(t)]\leq n.$ 2. (2)

If $a[e^{\lambda s}-R(t,s)]=n,$ then $a[\lambda-A(t)]\leq n.$

Proof.

1. (1)

Let $y\in Rg[\lambda-A(t)]^{n}$, then there exists $x\in\mathcal{D}^{n}$ (domain of $A(t)^{n}$) satisfying,

[TABLE]

Since $d[e^{\lambda s}-R(t,s)]=n,$ therefore $Rg[e^{\lambda s}-R(t,s)]^{n}=Rg[e^{\lambda s}-R(t,s)]^{n+1}.$ Hence, there exists $z\in X$ such that

[TABLE]

On the other hand, by theorem 2.2, we have,

[TABLE]

Thus we have,

[TABLE]

Therefore, we conclude that $y\in Rg[\lambda-A(t)]^{n+1}$ and hence,

[TABLE]

Finally, we conclude that

[TABLE] 2. (2)

Let $x\in N(\lambda-A(t))^{n+1}$ and we suppose that $a[e^{\lambda s}-R(t,s)]=n$, then we obtain

[TABLE]

From Corollary 2.1, we have

[TABLE]

hence

[TABLE]

Thus we have,

[TABLE]

Therefore, we obtain $x\in N(\lambda-A(t))^{n}$ and hence

[TABLE]

∎

Corollary 2.2**.**

Let $A(t)$ be a closed and densely defined generator of a $C_{0}-$quasi-semigroup $(R(t,s))_{t,s\geq 0}$ on a Banach space $X$. For all $\lambda\in\mathbb{C}$ and all $t,s\geq 0$, we have

1. (1)

$e^{\sigma_{a}(A(t))s}\subseteq\sigma_{a}(R(t,s));$** 2. (2)

$e^{\sigma_{d}(A(t))s}\subseteq\sigma_{d}(R(t,s)).$** 3. (3)

$e^{\sigma_{D}(A(t))s}\subseteq\sigma_{D}(R(t,s)).$**

Proof.

Immediately comes from propositions 2.2 and 2.1 . ∎

The following theorem examines the semi-Browder spectrum.

Theorem 2.4**.**

Let $A(t)$ be a closed and densely defined generator of a $C_{0}-$quasi-semigroup $(R(t,s))_{t,s\geq 0}$ on a Banach space $X$. For all $\lambda\in\mathbb{C}$ and all $t,s\geq 0$, we have

1. (1)

$e^{\sigma_{{Br}_{+}}(A(t))s}\subseteq\sigma_{{Br}_{+}}(R(t,s));$** 2. (2)

$e^{\sigma_{{Br}_{-}}(A(t))s}\subseteq\sigma_{{Br}_{-}}(R(t,s));$** 3. (3)

$e^{\sigma_{{Br}_{\pm}}(A(t))s}\subseteq\sigma_{{Br}_{\pm}}(R(t,s)).$**

Proof.

1. (1)

Suppose that $e^{\lambda s}\notin\sigma_{Br_{+}}(R(t,s))$, then there exist $n,m\in\mathbb{N}$ such that $\alpha[e^{\lambda s}-R(t,s)]=m$, $Rg[e^{\lambda s}-R(t,s)]$ is closed and $a[e^{\lambda s}-R(t,s)]=n.$ From Corollary 2.1 and Propositions 2.1 and 2.2, we obtain

$\alpha(\lambda-A(t))\leq m$, $Rg(\lambda-A(t))$ is closed and $a(\lambda-A(t))\leq n.$

Therefore $\lambda-A(t)\in\Phi_{+}(\mathcal{D})$ and $a(\lambda-A(t))<+\infty$ and hence, $\lambda\notin\sigma_{Br_{+}}(A).$ 2. (2)

Suppose that $e^{\lambda s}\notin\sigma_{Br_{-}}(R(t,s))$, then there exist $n,m\in\mathbb{N}$ such that $\beta[e^{\lambda s}-R(t,s)]=m$ and $d[e^{\lambda s}-R(t,s)]=n.$ By corollary 2.1 and Proposition 2.2, we obtain

$\beta(\lambda-A(t))\leq m$ and $d(\lambda-A(t))\leq n.$

Therefore $\lambda-A(t)\in\Phi_{-}(\mathcal{D})$ and $d(\lambda-A(t))<+\infty$ and hence, $\lambda\notin\sigma_{Br_{-}}(A(t)).$ 3. (3)

It is automatic by the previous assertions of this theorem.

∎

Proposition 2.3**.**

Let $A(t)$ be a closed and densely defined generator of a $C_{0}-$quasi-semigroup $(R(t,s))_{t,s\geq 0}$ on a Banach space $X$. For all $\lambda\in\mathbb{C}$ and all $t,s\geq 0$, we have

1. (1)

If $d_{e}[e^{\lambda s}-R(t,s)]=n,$ then $d_{e}[\lambda-A(t)]\leq n;$ 2. (2)

If $a_{e}[e^{\lambda s}-R(t,s)]=n,$ then $a_{e}[\lambda-A(t)]\leq n.$

Proof.

1. (1)

Suppose that $d_{e}[e^{\lambda s}-R(t,s)]=n$, Since $Rg[e^{\lambda s}-R(t,s)]^{n}\subseteq Rg(\lambda-A(t))^{n}$ we define the linear surjective application $\phi$ by

[TABLE]

Thus, by isomorphism Theorem, we obtain

[TABLE]

Therefore

[TABLE]

And since $N(\phi)\subseteq Rg[e^{\lambda s}-R(t,s)]^{n+1}\subseteq Rg(\lambda-A(t))^{n+1},$ then

[TABLE]

Finally, we obtain

[TABLE] 2. (2)

Suppose that

[TABLE]

And since $N(\lambda-A)^{n+1}\subseteq N[e^{\lambda s}-R(t,s)]^{n+1},$ we define the linear application $\psi$ by

[TABLE]

Thus, by isomorphism Theorem, we obtain

[TABLE]

Therefore

[TABLE]

And since $N(\psi)\subseteq N[e^{\lambda s}-R(t,s)]^{n}\subseteq Rg(\lambda-A(t))^{n},$ then

[TABLE]

Finally, we obtain

[TABLE]

∎

We will discuss in the following result the essential ascent and descent spectrum.

Theorem 2.5**.**

Let $A(t)$ be a closed and densely defined generator of a $C_{0}-$quasi-semigroup $(R(t,s))_{t,s\geq 0}$ on a Banach space $X$. For all $\lambda\in\mathbb{C}$ and all $t,s\geq 0$, we have

1. (1)

$e^{\sigma_{a_{e}}(A(t))s}\subseteq\sigma_{a_{e}}(R(t,s));$** 2. (2)

$e^{\sigma_{d_{e}}(A(t))s}\subseteq\sigma_{d_{e}}(R(t,s)).$**

Proof.

1. (1)

Suppose that , $e^{\lambda s}\notin\sigma_{a_{e}}(R(t,s)).$ Then there exists $n\in\mathbb{N}$ satisfying

[TABLE]

Therefore, by Proposition 2.3, we obtain $a_{e}[\lambda-A(t)]\leq n$ and hence

[TABLE] 2. (2)

Suppose that

[TABLE]

Then there exists $n\in\mathbb{N}$ satisfying

[TABLE]

Therefore, by Proposition 2.3, we obtain $d_{e}[\lambda-A(t)]\leq n$ and hence $\lambda\notin\sigma_{d_{e}}(A(t)).$

∎