Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators

• Published in: Linear & multilinear algebra Vol. 65; no. 6; pp. 1171 - 1193
• Main Authors: Zivkovic-Zlatanovic, Snezana C, Cvetkovic, Milos D
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 03-06-2017; Taylor & Francis Ltd
• Subjects: Algebra; approximate point (surjective) spectrum; Banach space; Decomposition; essential spectra; Kato operators; Linear operators; Operators; Operators (mathematics); Riesz operators; single-valued extension property; Subspaces
• ISSN: 0308-1087; 1563-5139
• DOI: 10.1080/03081087.2016.1231771

We shall say that a bounded linear operator T acting on a Banach space X admits a generalized Kato-Riesz decomposition if there exists a pair of T-invariant closed subspaces (M, N) such that
, the reduction
is Kato and
is Riesz. In this paper, we define and investigate the generalized Kato-Riesz spectrum of an operator. For T is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator S acting on X such that
,
,
is Riesz. We investigate generalized Drazin-Riesz invertible operators and also characterize bounded linear operators which can be expressed as a direct sum of a Riesz operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator. In particular, we characterize the single-valued extension property at a point
in the case that
admits a generalized Kato-Riesz decomposition.