Generalized Kato-Riesz decomposition

Abstract

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 X = M ⊕ N , the reduction TM is Kato and TN 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 T S = ST , ST S = S, T ST − T 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 λ0 ∈ C in the case that λ0 − T admits a generalized Kato-Riesz decomposition.