Property (w) for perturbations of polaroid operators

Property (R) for Bounded Linear Operators

“…This property, that we call property (R), means that the isolated points of the spectrum σ(T ) of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI − T is upper semi-Browder (see later for definitions). Property (R) is strictly related to a strong variant of classical Weyl's theorem, the so-called property (w) introduced by Rakočević in [26], and more extensively studied in recent papers ( [11], [3], [6], [8], [10]). We shall characterize property (R) in several ways and we shall also describe the relationships of it with the other variants of Weyl's theorem.…”

“…Property (w) and its perturbation properties has been studied in very recent papers ( [11], [6], [10], [3]). The following diagram resume the relationships between Weyl's theorems, a-Browder's theorem and property (w).…”

Quasinilpotent Part of w-Hyponormal Operators

“…Moreover, T is polaroid so also T K + is polaroid by Theorem 2.14 of [28]. By Theorem 2.10 of [26], then property ( )…”

“…T is an algebraically w -hyponormal then T has SVEP and hence T K + has SVEP by Theorem 2.14 of [28]. Moreover, T is polaroid so also T K + is polaroid by Theorem 2.14 of [28].…”

Tensor products and property (w)

“…Operators satisfying property (w) have been studied in a number of papers in the recent past, see [1][2][3] for additional references. It is known that an operator T satisfying property (w) satisfies W t, hence Bt, but the reverse implication is generally false; property (w) neither implies nor is implied by a-W t. Property (w) does not survive perturbations by commuting Riesz, even quasinilpotent, operators [1].…”