Weyl's Theorem for Class A Operators

Abstract

Abstract. In this paper, we show that Weyl's theorem holds for class A operators under a certain condition. We also show that a class A operator whose Weyl spectrum equals to the one-point set {0} is always compact and normal. (2000): 47A53, 47B20. Key words and phrases: Class A operators, w -hyponormal operators, continuity of spectra, Weyl's theorem. Mathematics subject classification R E F E R E N C E S

WEYL'S THEOREM AND TENSOR PRODUCT FOR OPERATORS SATISFYING T*k|T2|Tk≥T*k|T|2Tk

“…In [14], W. Y. Lee and S. H. Lee showed that if T ∈ B(H) is a hyponormal operator and f ∈ H(σ(T )), then Weyl's theorem holds for f (T ). Recently, this result was extended to p-quasihyponormal operators, class A operators and quasi-class A operators in [19], [18] and [5], respectively. In this section we show that if T ∈ B(H) is a quasi-class (A, k) operator and f ∈ H(σ(T )), then Weyl theorem holds for f (T ).…”

“…In [14], W. Y. Lee and S. H. Lee showed that if T is hyponormal operator, then Weyl's theorem holds for f (T ), where f is an analytic function on a neighborhood of spectrum of T . Recently, this result was extended to p-quasihyponormal operators, class A operators and quasi-class A operators in [19], [18] and [5], respectively.…”

On the Isolated Points of the Spectrum of Paranormal Operators

“…From these facts and Lemma 3.2. we have the following lemma. [8]) Let T be a paranormal operator and λ ∈ σ(T ) be an isolated point. Then the Riesz idempotent E with respect to λ defined by (1) satisfies ranE = ker(T − λ).…”

Weyl's theorem for algebraically k-quasiclass A operators

“…Berberian [6,7]. And this result was generalized for p-hyponormal operators by M. Chō, M. Iton and S.Ōshiro [9], for class A operators by A. Uchiyama [31], for algebraically hyponormal operators by Y.M. Han [22] and for algebraically paranormal operators by R.E.…”