Generalized Drazin-Riesz invertibility for operator matrices

Abstract

Let $$A\in {\mathscr {B}}(X)$$, $$B\in {\mathscr {B}}(Y)$$ and $$C\in {\mathscr {B}}(Y,X)$$ where X and Y are infinite dimensional Banach or Hilbert spaces. Let $$M_{C}=\begin{pmatrix} A &{} C \\ 0 &{} B \\ \end{pmatrix}$$ 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 $$\bigcap _{C\in {\mathscr {B}}(Y,X)}\sigma _{gDR}(M_{C})$$ will be investigated and its relation to $$\bigcap _{C\in {\mathscr {B}}(Y,X)}\sigma _{b}(M_{C})$$ will be studied, where $$\sigma _{gDR}(M_{C})$$ and $$\sigma _{b}(M_{C})$$ denote the generalized Drazin-Riesz spectrum and the Browder spectrum, respectively.