WEYL TYPE THEOREMS FOR ALGEBRIACALLY CLASS $p$-$wA(s,t)$ OPERATORS

In this paper, we study Weyl type theorems for $f(T)$, where $T$ is algebraically class $p$-$wA(s, t)$ operator with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$ and $f$ is an analytic function defined on an open neighborhood of the spectrum of $T$. Also we show that if $A , B^{*} \in B(\mathcal{H}) $ are class $p$-$wA(s, t)$ operators with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$,then generalized Weyl's theorem , a-Weyl's theorem, property $(w)$, property $(gw)$ and generalized a-Weyl's theorem holds for $f(d_{AB})$ for every $f \in H(\sigma(d_{AB})$, where $ d_{AB}$ denote the generalized derivation $\delta_{AB}:B(\mathcal{H})\rightarrow B(\mathcal{H})$ defined by $\delta_{AB}(X)=AX-XB$ or the elementary operator $\Delta_{AB}:B(\mathcal{H})\rightarrow B(\mathcal{H})$ defined by $\Delta_{AB}(X)=AXB-X$.

On Quasi-Class A Operators

Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let A;B be operators in B(H). In this paper we prove that if A is quasi-class A and B* is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A*X = XB*. We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f(A) satisfies generalized Weyl's theorem. Other related results are also given.

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].