Further characterizations of property (VΠ) and some applications

We carry out characterizations with techniques provided by the local spectral theory of bounded linear operators T ∈ L(X), X infinite dimensional complex Banach space, which verify property (VΠ) introduced by Sanabria et al. (Open Math. 16(1) (2018), 289-297). We also carry out the study for polaroid operators and Drazin invertible operators that verify the property mentioned above.

Polaroid operators and Weyl type theorems

Let [Formula: see text] be a [Formula: see text]-quasiposinormal operator on a complex Hilbert space [Formula: see text]. In this paper, we give basic properties for [Formula: see text] and we show that a [Formula: see text]-quasiposinormal operator [Formula: see text] is polaroid. We also prove that all Weyl type theorems (generalized or not) hold and are equivalent for [Formula: see text], where [Formula: see text] is an analytic function defined on a neighborhood of [Formula: see text].

Polaroid Operators with Svep and Perturbations of Property (Gaw)

In this paper we establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (gaw) holds. In this work, we consider commutative perturbations by algebraic operator and quasinilpotent operator for T ∈ B(X ) such that T * satisfies property (gaw). We prove that if A is an algebraic and T ∈ PS(X ) is such that AT = TA, then ƒ(T * + A*) satisfies property (gaw) for every ƒ ∈ Hc(σ(T + A)). Moreover, we show that if Q is a quasi-nilpotent operator and T ∈ PS(X ) is such that TQ = QT, then ƒ(T * + Q*) satisfies the property (gaw) for every ƒ ∈ Hc(σ(T +Q)). At the end of this paper, we apply the obtained results to a number of subclasses of PS(X ).

Generalized Weyl’s theorems for polaroid operators

In this paper we establish necessary and sufficient conditions on bounded linear operators for which generalized Weyl’s theorem, or generalized a-Weyl theorem, holds. We also consider generalized Weyl’s theorems in the framework of polaroid operators and obtain improvements of some results recently established in [20] and [29].