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.
ON WEYL'S THEOREM FOR QUASI-CLASS A OPERATORS
Let T be a bounded linear operator on a complex in- flnite dimensional Hilbert space H. We say that T is a quasi-class A operator if T ⁄ jT 2 jT ‚ T ⁄ jTj 2 T. In this paper we prove that if T is a quasi-class A operator and f is a function analytic on a neigh- borhood of the spectrum of T, then f(T) satisfles Weyl's theorem and f(T ⁄ ) satisfles a-Weyl's theorem.
ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION
Let A denote the class of bounded linear Hilbert space operators with the property that jA 2 j ‚ jAj 2 . In this paper we show that A-operators are flnitely ascensive and that, for non-zero operators A and B, A › B is in A if and only if A and B are in A. Also, it is shown that if A is an operator such that p(A) is in A for a non-trivial polynomial p, then Weyl's theorem holds for f(A), where f is a function analytic on an open neighborhood of the spectrum of A.