Generalized Drazin-Riesz Invertibility for Operators Matrices

TL;DR

This paper investigates conditions under which certain operator matrices are generalized Drazin-Riesz invertible and explores the spectra related to these operators in infinite-dimensional Banach or Hilbert spaces.

Contribution

It provides necessary and sufficient conditions for the generalized Drazin-Riesz invertibility of upper triangular operator matrices and analyzes their spectra across all possible off-diagonal operators.

Findings

Characterization of generalized Drazin-Riesz invertibility conditions.

Analysis of the intersection of spectra over all off-diagonal operators.

Relationship between generalized Drazin-Riesz spectrum and Browder spectrum.

Abstract

Let $A\in\mathcal{B}(X)$, $B\in\mathcal{B}(Y)$ and $C\in\mathcal{B}(Y,X)$ where $X$ and $Y$ are infinite Banach or Hilbert spaces. Let $M_{C}=\begin{pmatrix} A & C\cr 0 & B \end{pmatrix}$ be $2\times 2$ upper triangular operator matrix acting on $X\oplus Y$. In this paper, we consider some necessary and sufficient conditions for $M_{C}$ to be generalized Drazin-Riesz invertible. Furthermore, the set $\bigcap_{C\in \mathcal{B}(Y,X)}\sigma_{gDR}(M_{C})$ will be investigated and their relation between $\bigcap_{C\in \mathcal{B}(Y,X)}\sigma_{b}(M_{C})$ will be studied, where $\sigma_{gDR}(M_{C})$ and $\sigma_{b}(M_{C})$ denote the generalized Drazin-Riesz spectrum and the Browder spectrum, respectively.

Equations

$${\Phi }_{+}(X)$$

$${\Phi }_{-}(X)$$

$${\Phi }_{le}(X)$$

$${\Phi }_{re}(X)$$

$$\Phi (X)$$

$${\sigma }_{b_{+}}(T)$$

$${\sigma }_{b_{-}}(T)$$

$${\sigma }_b(T)$$

$${\sigma }_{lb}(T)$$

$${\sigma }_{rb}(T)$$

$$T_M\text{mbox}and{T}_N\text{mbox}areBrowder\iff T=T_M\oplus T_N\text{mbox}isBrowder;$$

$$T_M\text{mbox}and{T}_N\text{mbox}areRiesz\iff T=T_M\oplus T_N\text{mbox}isRiesz.$$

$${\sigma }_{gKR}(T)$$

$${\sigma }_{eK}(T)$$

$$TS=ST,STS=S,T-TST\text{mbox}isRiesz.$$

$$gDRM(X):=\{T\in \mathcal{B}(X):T=T_1\oplus T_2\text{mbox}where{T}_1\text{mbox}isboundedbelowand{T}_2\text{mbox}isRiesz\};$$

$$gDRQ(X):=\{T\in \mathcal{B}(X):T=T_1\oplus T_2\text{mbox}where{T}_1\text{mbox}issurjectiveand{T}_2\text{mbox}isRiesz\}.$$

$${\sigma }_{gDR}(T)=\{\lambda \in \mathbb{C}:T-\lambda I\text{mbox}isnotgeneralizedDrzain-Rieszinvertible\}.$$

$${\sigma }_{gDR}(T)={\sigma }_{gKR}(T)\cup \text{mbox}acc{\sigma }_b(T).$$

$$\bigcap _{C\in \mathcal{B}(K,H)}{\sigma }_b(M_C)={\sigma }_{b_{+}}(A)\cup {\sigma }_{b_{-}}(B)\cup W(A,B),$$

$$\bigcap _{C\in \mathcal{B}(K,H)}{\sigma }_{gDR}(M_C)=\bigcap _{C\in \mathcal{B}(K,H)}{\sigma }_b(M_C).$$

$$M_C-M_{C}S{M}_C=\left(\begin{array}{cc}A-AUA& L\\ 0& B-BVB\end{array}\right)$$

$$A-\lambda I=\left(\begin{array}{cc}{A}_1-\lambda I& 0\\ 0& A_2-\lambda I\end{array}\right)\text{mbox}andB-\lambda I=\left(\begin{array}{cc}{B}_1-\lambda I& 0\\ 0& B_2-\lambda I\end{array}\right).$$

$$\text{mbox}ind(A-\lambda I)=\text{mbox}ind(A_1-\lambda I)\text{ and }\text{mbox}ind(B-\lambda I)=\text{mbox}ind(B_1-\lambda I).$$

$$\text{mbox}ind(A_1)$$

$$=-(\alpha (B-{\lambda }_{0}I)-\beta (B-{\lambda }_{0}I))=-\text{mbox}ind(B-{\lambda }_{0}I)$$

$$=-\text{mbox}ind(B_1-{\lambda }_{0}I)=-\text{mbox}ind(B_1).$$

$$\left(\begin{array}{cc}{A}_1& C_1\\ 0& B_1\end{array}\right):H_1\oplus K_1\to H_1\oplus K_1\text{mbox}isBrowder.$$

$$M_{C_{H_1\oplus K_1}}=\left(\begin{array}{cc}{A}_1& C_1\\ 0& B_1\end{array}\right)\text{mbox}isBrowder\text{mbox}and{M}_{C_{H_2\oplus K_2}}=\left(\begin{array}{cc}{A}_2& 0\\ 0& B_2\end{array}\right)\text{mbox}isRiesz.$$

$$\text{mbox}acc{\sigma }_{lb}(A)\cup \text{mbox}acc{\sigma }_{rb}(B)\cup \mathcal{W}\subset \bigcap _{C\in \mathcal{B}(Y,X)}{\sigma }_{gDR}(M_C),$$

$$\text{mbox}acc{\sigma }_{b_{+}}(A)$$

$$\subset \text{mbox}acc{\sigma }_{b_{+}}(A)\cup \text{mbox}acc{\sigma }_{b_{-}}(B)\cup \mathcal{W}\cup {\sigma }_{gKR}(A)\cup {\sigma }_{gKR}(B).$$

$$\bigcap _{C\in \mathcal{B}(K,H)}{\sigma }_{gDR}(M_C)=\text{mbox}acc{\sigma }_{b_{+}}(A)\cup \text{mbox}acc{\sigma }_{b_{-}}(B)\cup \mathcal{W}.$$

$${\sigma }_{gKR}(T)\subseteq {\sigma }_{eK}(T)\subseteq {\sigma }_{b_{+}}(T)\cap {\sigma }_{b_{-}}(T).$$

$${\sigma }_{b_{+}}(U)$$

$${\sigma }_{b_{-}}(V)$$

Full text

1. Introduction and preliminaries

Let $X$ and $Y$ denote infinite dimensional complex Banach spaces and $\mathcal{B}(X,Y)$ denotes the algebra of all bounded linear operators from $X$ into $Y$. If $X=Y$ we write $\mathcal{B}(X)$ instead of $\mathcal{B}(X,X)$. For $T\in\mathcal{B}(X)$, we denote $\alpha(T)$ the dimension of the kernel $\mathcal{N}(T)$, $\beta(T)$ the codimension of the range $\mathcal{R}(T)$ and $\sigma(T)$ the spectrum of $T$. An operator $T\in\mathcal{B}(X)$ is called bounded below if $T$ is one to one and $\mathcal{R}(T)$ is closed. Recall that an operator $T\in\mathcal{B}(X)$ is Kato if $\mathcal{R}(T)$ is closed and $\mathcal{N}(T)\subset\mathcal{R}(T^{n})$, $n\in\mathbb{N}$. Two important class of Kato operators is given by the class of bounded below operators, and the class of surjective operators.

We can define the set of upper semi-Fredholm operators, the set of lower semi-Fredholm operators, the set of left semi-Fredholm operators, the set of right semi-Fredholm operators and the set of Fredholm operators respectively as following:

[TABLE]

The left semi-Fredholm spectrum is denoted by $\sigma_{le}(T)=\{\lambda\in\mathbb{C}:T-\lambda I\notin\Phi_{le}(X)\}$ and the right semi-Fredholm spectrum is denoted by $\sigma_{re}(T)=\{\lambda\in\mathbb{C}:T-\lambda I\notin\Phi_{re}(X)\}$. If $T\in\mathcal{B}(X)$ is upper semi-Fredholm or lower semi-Fredholm then the index of $T$ is defined as $\mbox{ind}(T):=\alpha(T)-\beta(T)$. Now, we sall recall that the ascent of an operator $T$ is given by $\mbox{asc}(T):=\mbox{inf}\{n\in\mathbb{N}:\mathcal{N}(T^{n})=\mathcal{N}(T^{n+1})\}$ and the descent of $T$ is given by $\mbox{des}(T):=\mbox{inf}\{n\in\mathbb{N}:\mathcal{R}(T^{n})=\mathcal{R}(T^{n+1})\}$, where $\mbox{inf}\>\emptyset=\infty$.

An operator $T\in\mathcal{B}(X)$ is upper semi-Browder if $T$ is upper semi-Fredholm and $\mbox{asc}(T)<\infty$. If $T$ is lower semi-Browder then $T$ is lower semi-Fredholm and $\mbox{des}(T)<\infty$. The operator $T\in\mathcal{B}(X)$ is Browder if $T$ is Fredholm with finite ascent and decent. The classes of operators defined above motivate the definition of the following spectra:

[TABLE]

Taking into account that $\sigma_{b_{+}}(T)=\sigma_{b_{-}}(T^{*})$ and $\sigma_{b_{-}}(T)=\sigma_{b_{+}}(T^{*})$ where $T^{*}$ denotes the dual of $T$. A relevant case is obtained if we assume that $T$ is left semi-Fredholm with finite ascent, in this case $T$ is called left semi-Browder. If $T$ is right semi-Fredholm operator with finite descent then $T$ is right semi-Browder. The left semi-Browder spectrum and the right semi-Browder spectrum of $T\in\mathcal{B}(X)$ are, respectively, denoted by:

[TABLE]

Let $M$ be a subspace of $X$ for which $T(M)\subset M$, $T\in\mathcal{B}(X)$ then $M$ is called $T$-invariant, we denote by $T_{M}$ the restriction of $T$ to $M$. We say that $T$ is completely reduced by the pair $(M,N)$ and we denote $(M,N)\in\mbox{Red}(T)$, if $M$ and $N$ are two closed $T$-invariant subspaces of $X$ such that $X=M\oplus N$. Clearly, for $(M,N)\in\mbox{Red}(T)$, the operator $T\in\mathcal{B}(X)$ is of the form $T=T_{M}\oplus T_{N}$ and we say that $T$ is a direct sum of $T_{M}$ and $T_{N}$.

We shall say that an operator $T\in\mathcal{B}(X)$ is Riesz, if $T-\lambda I\in\Phi(X)$ for all $\lambda\in\mathbb{C}\backslash\{0\}$. In [1, Theorem 3.111] it was proved that $T$ is Riesz if and only if $T-\lambda I\in\Phi(X)$ and $\mbox{ind}(T-\lambda I)=0$ for every $\lambda\in\mathbb{C}\backslash\{0\}$. If there exists a non-zero complex polynomial $P(z)$ such that $P(T)$ is Riesz then we say that $T\in\mathcal{B}(X)$ is polynomially Riesz, these operators have been discussed in [6] and [10].

It is useful to mention that for $T\in\mathcal{B}(X)$ and $(M,N)\in\mbox{Red}(T)$ ([10],[12]):

[TABLE]

[TABLE]

Recall that the concept of ”generalized Kato-Riesz decomposition” was defined in [9]. Namely, an operator $T\in\mathcal{B}(X)$ admits a generalized Kato-Riesz decomposition (abbreviated GKRD) if there exists a pair $(M,N)\in\mbox{Red}(T)$ such that $T_{M}$ is Kato and $T_{N}$ is Riesz. Additionally, if we assume in the definition above that $T_{N}$ is quasi-nilpotent and $N$ is finite-dimensional then $T$ is called essentially Kato. In this case, it is easy to see that $T_{N}$ is nilpotent since every quasi-nilpotent operator on a finite dimensional space is nilpotent. It should be noted that the class of semi-Fredholm operators belong to the class of essentially Kato operators (see for instance, [11, Theorem 16.21].

The generalized Kato-Riesz spectrum as well as the essentially Kato spectrum are given by:

[TABLE]

The class of generalized Drazin-Riesz invertible operators was first introduced by SC̆. Živković-Zlatanović and M D. Cvetković in [9] as follows: an operator $T\in\mathcal{B}(X)$ is said to be generalized Drazin-Riesz invertible if there exists $S\in\mathcal{B}(X)$ such that

[TABLE]

It is well known by [9, Theorem 2.3] that $T\in\mathcal{B}(X)$ is generalized Drazin-Riesz invertible if and only if there exists $(M,N)\in\mbox{Red}(T)$ such that $T_{M}$ is Browder and $T_{N}$ is Riesz. Moreover, the definition of the following set can also be found in [9]

[TABLE]

[TABLE]

Mainly, an operator $T\in\mathcal{B}(X)$ is called generalized Drazin-Riesz bounded below (resp, generalized Drazin-Riesz surjective) if $T\in gDRM(X)$ (resp, $T\in gDRQ(X)$). The generalized Drazin-Riesz spectrum is given by

[TABLE]

Among other things, Browder as well as Riesz operators are generalized Drazin-Riesz invertible. Furthermore, $\sigma_{gKR}(T)$ and $\sigma_{gDR}(T)$ are compact subsets of the complex plane $\mathbb{C}$, possibly empty and the generalized Drazin-Riesz spectrum of $T\in\mathcal{B}(X)$ is characterized as follows:

[TABLE]

Hilbert space operators are also inserted among this paper. Accurately, $H$ and $K$ will be two infinite dimensional separable Hilbert spaces.

For $A\in\mathcal{B}(X)$, $B\in\mathcal{B}(Y)$ and $C\in\mathcal{B}(Y,X)$, we denote by $M_{C}$ the upper triangular operator matrix acting on $X\oplus Y$ of the form $M_{C}=\begin{pmatrix}A&C\\ 0&B\\ \end{pmatrix}.$

In the recent past, the study of upper triangular operator matrices was a matter of great interest. More importantly, if $A\in\mathcal{B}(H)$ and $B\in\mathcal{B}(K)$, the sets $\bigcap_{C\in\mathcal{B}(K,H)}\sigma_{*}(M_{C})$ were considered by many authors (see, [3], [4], [8]), where $\sigma_{*}$ runs over different kind of spectra. For instance, in [8] S.Zhang et al. have shown that

[TABLE]

where $W(A,B)=\{\lambda\in\mathbb{C}:\alpha(A-\lambda I)+\alpha(B-\lambda I)\neq\beta(A-\lambda I)+\beta(B-\lambda I)\}$.

The ultimate goal of this paper is to provide sufficient conditions for which $M_{C}$ is generalized Drazin-Riesz invertible. In this regard, if $A\in\mathcal{B}(H)$, $B\in\mathcal{B}(K)$ and the following conditions hold:

1. (i)

$A$ and $B$ both admit a generalized Kato-Riesz decomposition; 2. (ii)

the upper semi-Browder spectrum does not cluster at [math]; 3. (iii)

the lower semi-Browder spectrum does not cluster at [math]; 4. (iv)

there exists $\delta>0$ such that $\alpha(A-\lambda I)+\alpha(B-\lambda I)=\beta(A-\lambda I)+\beta(B-\lambda I)$ for every $0<|\lambda|<\delta$.

We prove then, the existence of $C\in\mathcal{B}(K,H)$ for which $M_{C}$ is generalized Drazin-Riesz invertible. Consequently, we obtain a description of $\bigcap_{C\in\mathcal{B}(K,H)}\sigma_{gDR}(M_{C})$. Additionally, we give necessary and sufficient condition for which

[TABLE]

2. Main results

The following lemmas are among the most wiedly used results of this paper. The first lemma is an overview of the punctured neighborhood theorem ([11, Theorem 18.7] and [1, page 35]).

Lemma 2.1**.**

[11]** If $T\in\Phi_{+}(X)\cup\Phi_{-}(X)$. Then there exists a constant $\varepsilon>0$ such that for all $\lambda\in D(0,\varepsilon)\backslash\{0\}$, $\alpha(\lambda I-T)$ and $\beta(\lambda I-T)$ are constant. Moreover, $\mbox{ind}(\lambda I-T)=\mbox{ind}(T)$ for every $|\lambda|<\varepsilon$.

Lemma 2.2**.**

[9]** Let $T\in\mathcal{B}(X)$.

1. (a)

The following assertions are equivalent:

1. (i)

$T$* admits a GKRD and $0\notin\mbox{acc}\sigma_{b_{+}}(T)$,* 2. (ii)

$T=T_{M}\oplus T_{N}$* with $T_{M}$ is upper semi-Browder and $T_{N}$ is Riesz.* 2. (b)

The following assertions are equivalent:

1. (i)

$T$* admits a GKRD and $0\notin\mbox{acc}\sigma_{b_{-}}(T)$,* 2. (ii)

$T=T_{M}\oplus T_{N}$* with $T_{M}$ is lower semi-Browder and $T_{N}$ is Riesz.*

Lemma 2.3**.**

Let $A\in\mathcal{B}(X)$, $B\in\mathcal{B}(Y)$ and $C\in\mathcal{B}(Y,X)$. If $M_{C}$ is generalized Drazin-Riesz invertible, then the following statements hold:

1. (i)

$0\notin\mbox{acc}\sigma_{lb}(A)$, 2. (ii)

$0\notin\mbox{acc}\sigma_{rb}(B)$, 3. (iii)

there exists $\delta>0$ such that $\alpha(\lambda I-A)+\alpha(\lambda I-B)=\beta(\lambda I-A)+\beta(\lambda I-B)$ for every $0<|\lambda|<\delta$.

Proof.

Certainly, $\sigma_{gDR}(M_{C})=\sigma_{gKR}(M_{C})\cup\mbox{acc}\sigma_{b}(M_{C}).$ Since $M_{C}$ is generalized Drazin-Riesz invertible then $0\notin\sigma_{gDR}(M_{C})$, it follows that $0\notin\mbox{acc}\sigma_{b}(M_{C})$, which implying that there exists $\delta>0$ such that $M_{C}-\lambda I$ is Browder for every $0<|\lambda|<\delta$. According to [8, Theorem 2.9] $A-\lambda I$ is left semi-Browder $B-\lambda I$ is right semi-Browder and $\alpha(\lambda I-A)+\alpha(\lambda I-B)=\beta(\lambda I-A)+\beta(\lambda I-B)$ for every $0<|\lambda|<\delta$. This proved that $0\notin\mbox{acc}\sigma_{lb}(A)$, $0\notin\mbox{acc}\sigma_{rb}(B)$ and $\alpha(\lambda I-A)+\alpha(\lambda I-B)=\beta(\lambda I-A)+\beta(\lambda I-B)$ for every $0<|\lambda|<\delta$. ∎

Proposition 2.1**.**

Let $A\in\mathcal{B}(X)$, $B\in\mathcal{B}(Y)$ and $C\in\mathcal{B}(Y,X)$. If $M_{C}$ is generalized Drazin-Riesz invertible with a Drazin-Riesz inverse $S$ of the form $\begin{pmatrix}U&W\\ 0&V\\ \end{pmatrix}$ then $A$ and $B$ are generalized Drazin-Riesz invertible.

Proof.

Let $S=\begin{pmatrix}U&W\\ 0&V\\ \end{pmatrix}$ be the generalized Drazin-Riesz inverse of $M_{C}$. It is easily seen that $AU=UA$, $BV=VB$, $UAU=U$ and $VBV=V$ since $M_{C}S=SM_{C}$ and $SM_{C}S=S$. On the other hand

[TABLE]

where $L=C-AUC-AWB-CVB$, we have $M_{C}-M_{C}SM_{C}$ is Riesz and hence, $\begin{pmatrix}A-AUA-\lambda I&L\\ 0&B-BVB-\lambda I\\ \end{pmatrix}$ is Fredholm for every $\lambda\in\mathbb{C}\backslash\{0\}$. Then by [4, Theorem 3.2], $\sigma_{le}(A-AUA)=\{0\}$ and $\sigma_{re}(B-BVB)=\{0\}$. Therefore, using [10, Theorem 2.3] we have $\sigma_{e}(A-AUA)=\{0\}$ and $\sigma_{e}(B-BVB)=\{0\}$. As a result, $A$ and $B$ are generalized Drazin-Riesz invertible. ∎

This theorem outlines sufficient conditions for which $M_{C}$ is generalized Drazin-Riesz invertible for some $C\in\mathcal{B}(K,H)$.

Theorem 2.1**.**

Let $A\in\mathcal{B}(H)$ and $B\in\mathcal{B}(K)$ such that the following statements hold:

1. (i)

$A$* and $B$ both admit a generalized Kato-Riesz decomposition,* 2. (ii)

$0\notin\mbox{acc}\sigma_{b_{+}}(A)$, 3. (iii)

$0\notin\mbox{acc}\sigma_{b_{-}}(B)$, 4. (iv)

there exists $\delta>0$ such that $\alpha(\lambda I-A)+\alpha(\lambda I-B)=\beta(\lambda I-A)+\beta(\lambda I-B)$ for every $0<|\lambda|<\delta$.

Then $M_{C}$ is generalized Drazin-Riesz invertible for some $C\in\mathcal{B}(K,H)$.

Proof.

By means of lemma 2.2, there exists a pair $(H_{1},H_{2})\in\mbox{Red}(A)$ such that $A_{H_{1}}=A_{1}$ is upper semi-Browder and $A_{H_{2}}=A_{2}$ is Riesz. Also, there exists a pair $(K_{1},K_{2})\in\mbox{Red}(B)$ such that $B_{K_{1}}=B_{1}$ is lower semi-Browder and $B_{K_{2}}=B_{2}$ is Riesz. It is clear that for each $\lambda\in\mathbb{C}\backslash\{0\}$, $\mbox{ind}(A_{2}-\lambda I)=0$ and $\mbox{ind}(B_{2}-\lambda I)=0$. Furthermore, $A_{1}$ is upper semi-Fredholm and $B_{1}$ is lower semi-Fredholm. So according to lemma 2.1 (punctured neighborhood theorem) there exists $\varepsilon>0$ such that $\mbox{ind}(A_{1})=\mbox{ind}(A_{1}-\lambda I)$ and $\mbox{ind}(B_{1})=\mbox{ind}(B_{1}-\lambda I)$ for every $|\lambda|<0$. It is straightforward that

[TABLE]

Thus, for each $0<|\lambda|<\varepsilon$

[TABLE]

Combining 1 with the statement (iv), we have for $\lambda_{0}\in\mathbb{C}$, $0<|\lambda_{0}|<\mbox{min}\{\varepsilon,\delta\}$.

[TABLE]

From [8, Theorem 2.9] it follows that there exists $C_{1}\in\mathcal{B}(K_{1},H_{1})$ such that:

[TABLE]

Define $C\in\mathcal{B}(K,H)$ as follows $\begin{pmatrix}C_{1}&0\\ 0&0\\ \end{pmatrix}:K_{1}\oplus K_{2}\rightarrow H_{1}\oplus H_{2}$. It is easy to show that $H_{1}\oplus K_{1}$ and $H_{2}\oplus K_{2}$ are closed subspaces of $H\oplus K$ invariant by $M_{C}=\begin{pmatrix}A&C\\ 0&B\\ \end{pmatrix}$. Moreover,

[TABLE]

Hence $M_{C}=\begin{pmatrix}A_{1}&C_{1}&0&0\\ 0&B_{1}&0&0\\ 0&0&A_{2}&0\\ 0&0&0&B_{2}\\ \end{pmatrix}:H_{1}\oplus K_{1}\oplus H_{2}\oplus K_{2}\rightarrow H_{1}\oplus K_{1}\oplus H_{2}\oplus K_{2}$ is generalized Drazin-Riesz invertible.

∎

Corollary 2.1**.**

1. (i)

Let $A\in\mathcal{B}(X)$ and $B\in\mathcal{B}(Y)$. Then:

[TABLE]

where $\mathcal{W}=\{\lambda\in\mathbb{C}:\nexists\delta>0\mbox{ such that }\alpha(A-\lambda I-\lambda^{\prime}I)+\alpha(B-\lambda I-\lambda^{\prime}I)=\beta(A-\lambda I-\lambda^{\prime}I)+\beta(B-\lambda I-\lambda^{\prime}I)\mbox{ for each }0<|\lambda^{\prime}|<\delta\}.$ 2. (ii)

Let $A\in\mathcal{B}(H)$ and $B\in\mathcal{B}(K)$. Then:

[TABLE]

In particular, if $\sigma_{gKR}(A)\subseteq\mbox{acc}\sigma_{b_{+}}(A)$ and $\sigma_{gKR}(B)\subseteq\mbox{acc}\sigma_{b_{-}}(B)$ then

[TABLE]

Proof.

Due to lemma 2.3 and theorem 2.1 . ∎

Remarks 1**.**

Let $T\in\mathcal{B}(X)$

1. (a)

If $T$ is either upper semi-Browder or lower semi-Browder then $T$ is essentially Kato and hence admits a GKRD. Consequently, the following inclusions hold:

[TABLE]

It follows that $\sigma_{gKR}(T)\subseteq acc\sigma_{b_{+}}(T)\cap acc\sigma_{b_{-}}(T)$ when $\sigma_{b_{+}}(T)=acc\sigma_{b_{+}}(T)$ and $\sigma_{b_{-}}(T)=acc\sigma_{b_{-}}(T)$. 2. (b)

If $T$ is polynomially Riesz then $\sigma_{gKR}(T)=acc\sigma_{b_{+}}(T)=acc\sigma_{b_{-}}(T)=\emptyset$.

The following example is coted from [3], it is also valid to build a particular operator matrix who satisfies the conditions of theorem 2.1 .

Example 1**.**

$U$* and $V$ are the forward and the backward unilateral shifts on $l^{2}(\mathbb{N})$, we have:*

[TABLE]

*where $\mathbb{D}=\{\lambda\in\mathbb{C}:|\lambda|\leqslant 1\}$. It is known that $U$ is upper semi-Browder, $V$ is lower semi-Browder with $\mbox{asc}(U)=\mbox{des}(V)=1$, $\mbox{ind}(U)=-1$ and $\mbox{ind}(V)=1$. Following remark 1, $U$ and $V$ admit a GKRD.

Now let $W\in\mathcal{B}(l^{2}(\mathbb{N})\oplus l^{2}(\mathbb{N}))$ defined by $W=\begin{pmatrix}V&0\\ 0&0\\ \end{pmatrix}$. Then $\sigma_{b_{-}}(W)=\partial\mathbb{D}\cup\{0\}$ and $0\notin acc\sigma_{b_{-}}(W)$, but $W$ admits a GKRD.

On the other hand, from lemma 2.1 there exists $\delta>0$ such that $\mbox{ind}(U-\lambda I)=\mbox{ind}(U)$ and $\mbox{ind}(V-\lambda I)=\mbox{ind}(V)$ for $0<|\lambda|<\delta$, then:*

[TABLE]

for every $0<|\lambda|<\delta$. Thus by theorem 2.1 there exists $C\in\mathcal{B}(l^{2}(\mathbb{N})\oplus l^{2}(\mathbb{N}),l^{2}(\mathbb{N}))$ such that:

[TABLE]

is generalized Drazin-Riesz invertible, i.e. $0\notin\bigcap_{C\in\mathcal{B}(l^{2}(\mathbb{N})\oplus l^{2}(\mathbb{N}),l^{2}(\mathbb{N}))}\sigma_{gDR}(M_{C})$.

In connection with Theorem 2.1, it is straightforward to obtain the following result.

Proposition 2.2**.**

Let $A\in\mathcal{B}(H)$ and $B\in\mathcal{B}(K)$ such that the following statments hold:

1. (i)

$A$* is generalized Drazin-Riesz bounded below,* 2. (ii)

$B$* is generalized Drazin-Riesz surjective,* 3. (iii)

there exists $\delta>0$ such that $\alpha(\lambda I-A)+\alpha(\lambda I-B)=\beta(\lambda I-A)+\beta(\lambda I-B)$ for every $0<|\lambda|<\delta$.

Then there exists $C\in\mathcal{B}(K,H)$ such that $M_{C}$ is generalized Drazin-Riesz invertible.

Proof.

Because of [9, Theorem 2.4] and [9, Theorem 2.5] the following equivalences hold:

[TABLE]

Therefore, by the proof of Theorem 2.1 there exists $C\in\mathcal{B}(K,H)$ such that:

[TABLE]

[TABLE]

As a result, $M_{C}$ is generalized Drazin-Riesz invertible.

∎

Let $K\subseteq\mathbb{C}$ be a compact set. Denote by $\eta K$ the connected hull of $K$, where the complement $\mathbb{C}\backslash\eta K$ is the unique unbounded component of the complement $\mathbb{C}\backslash K$ ([7, Definition 7.10.1]), a hole of $K$ is a component of $\eta K\backslash K$.

If $H,K\subseteq\mathbb{C}$ are compact subsets, we get ([7, Theorem 7.10.3]):

[TABLE]

For instance, if $K\subseteq\mathbb{C}$ is finite then $\eta K=K$. Moreover the following implication holds:

[TABLE]

and in this case $H=K$.

Theorem 2.2**.**

For $A\in\mathcal{B}(X)$, $B\in\mathcal{B}(Y)$ and $C\in\mathcal{B}(Y,X)$. Then the following statements are equivalent:

1. (i)

$\sigma_{gKR}(M_{C})=\emptyset$, 2. (ii)

$\sigma_{gDR}(M_{C})=\emptyset$, 3. (iii)

$A$* and $B$ are polynomially Riesz,* 4. (iv)

$\sigma_{b}(M_{C})$* is finite.*

Proof.

$(i)\Leftrightarrow(ii)$ From [9, Theorem 3.10], it follows that $\eta\sigma_{gKR}(M_{C})=\eta\sigma_{gDR}(M_{C})$. This implies that $\sigma_{gKR}(M_{C})=\emptyset$ if and only if $\sigma_{gDR}(M_{C})=\emptyset$.

$(ii)\Rightarrow(iv)$ assume that $\sigma_{gDR}(M_{C})=\emptyset$, since $\sigma_{gDR}(M_{C})=\sigma_{gKR}(M_{C})\cup\mbox{acc}\sigma_{b}(M_{C})$. then $\mbox{acc}\sigma_{b}(M_{C})=\emptyset$. Consequently, $\sigma_{b}(M_{C})$ is a finite set.

$(iii)\Rightarrow(ii)$ Let $A$ and $B$ be polynomially Riesz, then $\sigma_{b}(A)=\pi_{A}^{-1}(0)=\{\lambda_{1},\lambda_{2},...,\lambda_{n}\}$ and $\sigma_{b}(B)=\pi_{B}^{-1}(0)=\{\mu_{1},\mu_{2},...,\mu_{m}\}$ where $\pi_{A}$ and $\pi_{B}$ are the minimal polynomials of $A$ and $B$. According to the proof of [12, Theorem 2.6] we get,

[TABLE]

Also, it is known that $\sigma_{b}(M_{C})\subset\sigma_{b}(A)\cup\sigma_{b}(B)$. Since $\sigma_{b}(M_{C})$ and $\sigma_{b}(A)\cup\sigma_{b}(B)$ are finite then, $\sigma_{b}(M_{C})=\sigma_{b}(A)\cup\sigma_{b}(B)=\pi_{A}^{-1}(0)\cup\pi_{B}^{-1}(0)=\{\lambda_{1},\lambda_{2},...,\lambda_{n},\mu_{1},\mu_{2},...,\mu_{m}\}.$ Thus, for every $\lambda\notin\pi_{A}^{-1}(0)\cup\pi_{B}^{-1}(0)$, $M_{C}-\lambda$ is Browder and hence generalized Drazin-Riesz invertible.

Now from [10, Theorem 2.13] the Banach spaces $X$ and $Y$ are decomposed into the direct sums $X=X_{1}\oplus X_{2}\oplus...\oplus X_{n},$ $Y=Y_{1}\oplus Y_{2}\oplus...\oplus Y_{m}$ where $X_{i}$ is closed $A$-invariant subspace of $X$ (resp, $Y_{j}$ is closed $B$-invariant subspace of $Y$). $A=A_{1}\oplus A_{2}\oplus...\oplus A_{n}$ and $B=B_{1}\oplus B_{2}\oplus...\oplus B_{m}$, $A_{i}=A_{X_{i}}$ and $B_{j}=B_{Y_{j}}$.

$A_{i}-\lambda_{i}$ and $B_{j}-\mu_{j}$ are Riesz which implies that $\sigma_{b}(A_{i})\subset\{\lambda_{i}\}$ and $\sigma_{b}(B_{j})\subset\{\mu_{j}\}$ for i=1,…,n ; j=1,…,m.

Case 1: If $n=m$, then $A_{i}-\lambda_{j}$ and $B_{i}-\mu_{j}$ are Browder for every $i\neq j$, $i,j\in\{1,...,n\}.$ It is clear that $X_{1}\oplus Y_{1}$, $X_{2}\oplus Y_{2}$, … ,$X_{n}\oplus Y_{n}$ are closed subspace of $X\oplus Y$ invariant by $M_{C}$. Moreover, $\sigma_{b}({M_{C}}_{X_{i}\oplus Y_{i}})=\sigma_{b}(A_{i})\cup\sigma_{b}(B_{i})\subset\{\lambda_{i},\mu_{i}\}$ for $i=1,...,n$. Using the decomposition:

[TABLE]

We have $M_{C}-\lambda_{1}$ is Browder, because ${M_{C}}_{X_{i}\oplus Y_{i}}-\lambda_{1}$ are Browder for every $i\in\{1,2,...,n\}$. It follows that $M_{C}-\lambda_{1}$ is generalized Drazin-Riesz invertible. In the same way, we get that $M_{C}-\lambda_{i}$ and $M_{C}-\mu_{i}$ are generalized Drazin-Riesz invertible for every $i\in\{1,...,n\}$.

Case 2: If $n\neq m$ we will consider only the case where $n>m$ since the argument of the case $n<m$ is similar. So, $M_{C}-\lambda_{i}$ and $M_{C}-\mu_{i}$ are Browder thus, $M_{C}-\lambda_{i}$ and $M_{C}-\mu_{i}$ are generalized Drazin-Riesz invertible for every $i\in\{1,...,m\}$. For $i=m+1$, we consider the decomposition:

[TABLE]

Since ${M_{C}}_{X_{m+1}}-\lambda_{m+1}$ is Riesz and ${M_{C}}_{X_{1}\oplus Y_{1}}-\lambda_{m+1}\oplus{M_{C}}_{X_{2}\oplus Y_{2}}-\lambda_{m+1}\oplus...\oplus{M_{C}}_{X_{m}\oplus Y_{m}}-\lambda_{m+1}\oplus{M_{C}}_{X_{m+2}}-\lambda_{m+1}\oplus...\oplus{M_{C}}_{X_{n}}-\lambda_{m+1}$ is Browder. As a result, $M_{C}-\lambda_{m+1}$ is generalized Drazin-Riesz invertible. Similarly, we can prove that $M_{C}-\lambda_{k}$ is generalized Drazin-Riesz invertible for every $k\in\{m+2,...,n\}$.

Which shows that $M_{C}-\lambda$ is generalized Drazin-Riesz invertible for every $\lambda\in\mathbb{C}$. consequently, $\sigma_{gDR}(M_{C})=\emptyset$.

$(iii)\Rightarrow(iv)$ obvious.

$(iv)\Rightarrow(iii)$ suppose that $\sigma_{b}(M_{C})$ is finite, we have

[TABLE]

Thus $\sigma_{b}(M_{C})=\sigma_{b}(A)\cup\sigma_{b}(B)$ which implies that $\sigma_{b}(A)$and $\sigma_{b}(B)$ are finite. Then according to [10, Theorem 2.3] $A$ and $B$ are polynomially Riesz.

∎

Lemma 2.4**.**

Let $T\in\mathcal{B}(X)$.

1. (i)

$T$* is essentially Kato and $0\notin\mbox{acc}\sigma_{b_{+}}(T)$ if and only if $T$ is upper semi-Browder.* 2. (ii)

$T$* is essentially Kato and $0\notin\mbox{acc}\sigma_{b_{-}}(T)$ if and only if $T$ is lower semi-Browder.*

Proof.

$(i)$ If $T$ is upper semi Browder then $T$ is essentially Kato. Since $0\notin\sigma_{b_{{}_{+}}}(T)$ then $0\notin\mbox{acc}\sigma_{b_{+}}(T)$.

Conversely, Suppose that $T$ is essentially Kato and $0\notin\mbox{acc}\sigma_{b_{+}}(T)$ then, by [5, Theorem 2.2], there exists $\delta>0$ such that $T-\lambda$ is Kato for each $\lambda$ such that $0<|\lambda|<\delta$. Since $0\notin\mbox{acc}\sigma_{b_{+}}(T)$ there exists $\delta^{\prime}>0$ such that $T-\lambda$ is upper semi-Browder for every $0<|\lambda|<\delta^{\prime}$. For the seek of simplicity we tend to assume that $\delta=\delta^{\prime}$. Hence, from [11, Lemma 20.9], $T-\lambda$ is bounded below for every $0<|\lambda|<\delta$, which means that $0\notin\mbox{acc}\sigma_{ap}(T)$. Moreover, let $(M,N)\in\mbox{Red}(T)$ where $T_{M}$ is Kato, $T_{N}$ is nilpotent and $N$ is finite dimensional. Clearly,

[TABLE]

As $0\notin\mbox{acc}\sigma_{ap}(T)$, it follows that $0\notin\mbox{acc}\sigma_{ap}(T_{M})$ also $T_{M}$ is Kato then according to [2, theorem 2.14] $T_{M}$ is bounded below. Altogether, $T$ is upper semi-Browder [11, Theorem 20.10].

$(ii)$ Suppose that $T$ is essentially Kato then, by [11, Theorem 21.5], $T^{*}$ is essentially Kato. Also, we have $0\notin\mbox{acc}\sigma_{b_{-}}(T)=\mbox{acc}\sigma_{b_{+}}(T^{*})$ and hence, by the first part $T^{*}$ is upper semi-Browder, or equivalently $T$ is lower semi-Browder.

The reverse implication is obvious. ∎

Under further hypothesis the converse of theorem 2.1 is also hold.

Proposition 2.3**.**

Let $A\in\mathcal{B}(H)$ and $B\in\mathcal{B}(K)$ be essentially Kato operators, if $M_{C}$ is generalized Drazin-Riesz invertible for some $C\in\mathcal{B}(K,H)$. Then $A$ is upper semi-Browder, $B$ is lower semi-Browder and $\alpha(A)+\alpha(B)=\beta(A)+\beta(B)$ i.e. $M_{C_{1}}$ is Browder for some $C_{1}\in\mathcal{B}(K,H)$.

Proof.

It follows from lemma 2.3 that $0\notin\mbox{acc}\sigma_{b_{+}}(A)\cup\mbox{acc}\sigma_{b_{-}}(B)$ and there exists $\delta>0$ such that $\alpha(\lambda I-A)+\alpha(\lambda I-B)=\beta(\lambda I-A)+\beta(\lambda I-B)$ for every $0<|\lambda|<\delta$. In addition to this $A$ is upper semi-Browder and $B$ is lower semi-Browder according to lemme 2.4. Also, by lemma 2.1 it is easy to see that $\alpha(A)+\alpha(B)=\beta(A)+\beta(B)$. Thus, following [8, Theorem 2.9] there exists $C_{1}\in\mathcal{B}(K,H)$ such that $M_{C_{1}}$ is Browder. ∎

Corollary 2.2**.**

Let $A\in\mathcal{B}(H)$ and $B\in\mathcal{B}(K)$. Then:

[TABLE]

Corollary 2.3**.**

Let $A\in\mathcal{B}(H)$ and $B\in\mathcal{B}(K)$. Then the following statements are equivalent:

1. (i)

$\bigcap_{C\in\mathcal{B}(K,H)}\sigma_{gDR}(M_{C})=\bigcap_{C\in\mathcal{B}(K,H)}\sigma_{b}(M_{C})$, 2. (ii)

$\sigma_{eK}(A)\cup\sigma_{eK}(B)\subset\bigcap_{C\in\mathcal{B}(K,H)}\sigma_{gDR}(M_{C})$.