A Note on Drazin Invertibility for Upper Triangular Block Operators

Abstract

A bounded linear operator A acting on a Banach space X is said to be an upper triangular block operators of order n, and we write \({A \in \mathcal{UT}_{n}(X)}\) , if there exists a decomposition of \({X = X_{1} \oplus . . . \oplus X_{n}}\) and an n × n matrix operator \({(A_{i,j})_{\rm 1 \leq i, j \leq n}}\) such that \({A = (A_{i, j})_{1 \leq i, j \leq n}, A_{i, j} = 0}\) for i > j. In this note we characterize a large set of entries Ai, j with j > i such that \({\sigma_{\rm D} (A) = {\bigcup\limits_{i = 1}^{n}} \sigma_{\rm D} (A_{i, i})}\) ; where σD(.) is the Drazin spectrum. Some applications concerning the Fredholm theory and meromorphic operators are given.