Property (R) for Bounded Linear Operators

Abstract

We introduce the spectral property (R), for bounded linear operators defined on a Banach space, which is related to Weyl type theorems. This property is also studied in the framework of polaroid ,or left polaroid, operators.Mathematics Subject Classification (2010). Primary 47A10, 47A11. Secondary 47A53, 47A55.

Properties (S) and (gS) for bounded linear operators

“…In this last part, we give a summary of the known Weyl type theorems as in [11], including the properties introduced in [4,6,14,24,28], and in this paper. We use the abbreviations gaW, aW, gW, W, (gw), (w), (Bw), In the following diagram, which extends the similar diagram presented in [28], arrows signify implications between various Weyl type theorems, Browder type theorems, property (gw), property (gb), property (Bw), property (Bgw), property (Bb), property (Bgb), property (R), property (gR), property (S) and property (gS).…”

“…It is shown in Theorem 2.3 of [14] that an operator possessing property (gb) possesses property (b) but the converse is not true in general, see also [26]. Following [4], we say an operator T ∈ L (X ) is said to be satisfies property (R) if π 0 a (T ) = E 0 (T ). In Theorem 2.4 of [4], it is shown that T satisfies property (w) if and only if T satisfies a-Browder's theorem and T satisfies property (R).…”

“…Following [4], we say an operator T ∈ L (X ) is said to be satisfies property (R) if π 0 a (T ) = E 0 (T ). In Theorem 2.4 of [4], it is shown that T satisfies property (w) if and only if T satisfies a-Browder's theorem and T satisfies property (R).…”

A new variation of Weyl type theorems and perturbations

“…Recall [6,8] that property (aw) is said to hold for T if ∆(T) = π a 00 (T), and property (R) holds for T if p a 00 (T) = π 00 (T). The single valued extension property plays an important role in local spectral theory, see the recent monograph of Laursen and Neumamn [13] and Aiena [2].…”

Isolated eigenvalues, poles and compact perturbations of Banach space operators

“…Examples of analytic Toeplitz operators and operators satisfying the abstract shift condition are considered. (P 1) : E(A) = Π a (A) and (P 2) : E a (A) = Π(A)and their stability under perturbations by commuting Riesz operators, have been studied in a number of papers in the recent past, amongst them [2,3,4,8,9,17,18,20,21].…”