Generalized Drazin invertibility and local spectral theory
In the present paper, the author generalizes the results of \textit{P. Aiena} and \textit{S. Triolo} [J. Math. Anal. Appl. 435, No. 1, 414--424 (2016; Zbl 1339.47006)]. The author proves that, if \(A\in B(\mathcal{X})\) is an upper triangular Banach space operator with diagonal \((A_{1},A_{2})\), \(A_{1}\) is invertible and \(A_{2}\) is quasinilpotent, then \(A^{-1}_{1}\oplus A_{2}\) satisfies either of the single valued extension property, Dunford's condition (C), Bishop's property \((\beta)\), decomposition property \(\delta\) or is decomposable if and only if \(A_{1}\) has the respective property. The operator \(A^{-1}_{1}\oplus0\) is subscalar, left polaroid, right polaroid if and only if \(A_{1}\) is. As a consequence, the author deduces that, for the Drazin invertible operators \(A,\) with Drazin inverse \(B\), the operator \(B\) satisfies any one of the precedent properties if and only if \(A\) does.
local spectral properties (C), \((\beta)\) and \(\delta\)
Eschmeier-Putinar-Bishop property \((\beta)\)
scientific article; zbMATH DE number 6704120
scientific article