On the Poles of the Local Resolvent

Abstract

A b s t r a c t .We give two characterizations of the isolated singularities of the local resolvent function of an operator T E L ( X ) at a point z of a complex Banach space X: in terms of a suitable decomposition of x, and in terms of the existence of a sequence in X related with the Laurent series of the local resolvent function. Moreover, we introduce the locally chain-finite operators at a pointx E X and show that T is chain-finite if and only if T is locally chain-finite at every x E X.

Local Ergodic Theorems for C0-Semigroups

“…The following results have been proven by T. BERMÚDEZ and al in [3]. Actually, we can easily check that these results are also satisfactory when we choose A is a closed linear operator with domain D(A) ⊂ X.…”

“…We recall a characterization of the poles of the local resolvent function obtained in the monograph [3].…”

“…Remark 2.1. Let T ∈ B(X) has the SVEP and λ ∈ C. It is well known (see [3]) that if λ is a pole of local resolvent function xT for all x ∈ X does not imply that λ is a pole of the resolvent function of T . But, if we suppose further that {λ} σ A (x), the implication becomes true.…”

Chain-finite operators and locally chain-finite operators

“…Namely, given T P LX and x P X, we say that T is a locally chain-®nite operator at x with lT; x k > 0 if T kÀ1 x; T T T k x; T and with lT; x 0 if 0 = P x; T [4, De®nition 4.1]. This notion is a localization of the concept of chain-®nite operator: if T satis®es the Single Valued Extension Property (hereafter referred to as SVEP), then T is a chain-®nite operator if and only if T is a locally chain-®nite operator at x for every x P X [4,Theorem 4.2]. Moreover, locally chain-®nite operators have the properties that 0 is a pole of the local resolvent function and that the vector has a unique decomposition similar to (2).…”

“…equivalently, there exists a unique decomposition x x 1 x 2 such that x 1 P NT k n NT kÀ1 and x 2 ; T x; T n f0g by [4,Theorem 3.3]. For convenience, we shall say that 0 is a pole of the local resolvent function of T at x of order 0 if 0 P x; T .…”