Generalized Drazin-Riesz invertibility for operator matrices (Q1989141)

Let \(X\) be a complex Banach space and \(\mathcal{B}(X)\) denote the space of all bounded linear operators on \(X.\) \(T\in\mathcal{B}(X)\) is called a \textit{Riesz} operator if, for all \(\lambda \in \mathbb{C} \setminus \{0\}\), the operator \(T-\lambda I\) has a finite dimensional kernel, the range space \(R(T-\lambda I)\) is closed, and has finite codimension. \(T\in\mathcal{B}(X)\) is said to be \textit{generalized Drazin-Riesz invertible} if there exists \(S\in\mathcal{B}(X)\) such that \(TS=ST, STS=S\), and \(T-TST\) is a Riesz operator. The authors obtain, among other results, necessary and sufficient conditions for a block upper triangular operator matrix defined on the direct sum of two Banach or Hilbert spaces to be generalized Drazin-Riesz invertible.