Generalized Drazin-Riesz Invertibility for Operators Matrices

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 $M_C=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)$ 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.