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 an elementary operator with 2-isometric operator entries

A Hilbert space operator T is said to be a 2-isometric operator if T*2T2- 2T*T + I = 0. Let dAB ? B(B(H)) denote either the generalized derivation ?AB = LA-RB or the elementary operator ?= LARB-I, we show that if A and B* are 2-isometric operators, then, for all complex ?, (dAB-?)-1(0)? (d*AB-?)-1(0), the ascent of (dAB-?) ? 1, and dis polaroid. Let H(?(dAB)) denote the space of functions which are analytic on ?(dAB), and let Hc(?(dAB)) denote the space of f ? H(?(dAB)) which are non-constant on every connected component of ?(dAB), it is proved that if A and B* are 2-isometric operators, then f(dAB) satisfies the generalized Weyl?s theorem and f(d*AB) satisfies the generalized a-Weyl?s theorem.

On the invertibility of length two elementary operators

Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.

Additive maps having a generalized derivation expansion

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.