Property (ω) and quasi-class (A, k) operators
In this paper, we prove the following assertions: (i) If T is of quasi-class (A, k), then T is polaroid and reguloid; (ii) If T or T* is an algebraically of quasi-class (A, k) operator, then Weyls theorem holds for f(T) for every f ∈ Hol(σ(T)); (iii) If T* is an algebraically of quasi-class (A, k) operator, then a-Weyls theorem holds for f(T) for every f ∈ Hol(σ(T)); (iv) If T* is algebraically of quasi-class (A, k) then property (ω) holds for T.
Quasinilpotent Part of class A or (p, k)-quasihyponormal Operators
Let T be a bounded linear operator on a complex Hilbert space ℌ. T is called a class A operator if |T|2 ≤ |2|, and (p, k)-quasihyponormal if T* k ((T*T) p − (TT*) p )T k ≥ 0. In this paper, we characterize the quasinilpotent part ℌ 0(T − λ) = {x ∈ ℌ: \( \mathop {\lim }\limits_{n \to \infty } \left\| {(T - \lambda )^n x} \right\|^{\tfrac{1} {n}} = 0 \)} of a class A or (p, k)-quasihyponormal operator T.
Weyl’s theorem for totally hereditarily normaloid operators
A Banach space operatorT ∈B(χ) is said to behereditarily normaloid, denotedT ∈ ℋN, if every part ofT is normaloid;T ∈ ℋN istotally hereditarily normaloid, denotedT ∈ ℑHN, if every invertible part ofT is also normaloid. Class ℑHN is large; it contains a number of the commonly considered classes of operators. The operatorT isalgebraically totally hereditarily normaloid, denotedT ∈a — ℑHN, both non-constant polynomialp such thatp(T) ∈ ℑHN. For operatorsT ∈a − ℑHN, bothT andT* satisfy Weyl’s theorem; if also either ind(T−μ)≥0 or ind(T−μ)≤0 for all complexμ such thatT−μ is Fredholm, thenf(T) andf(T*) satisfy Weyl’s theorem for all analytic functionsf ∈ ℋ(σ(T)). For operatorsT ∈a — ℑHN such thatT has SVEP,T* satisfiesa-Weyl’s theorem.