# 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.

## Key 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.