Weyl's theorem for algebraically k-quasiclass a operators

Tyt. z nagłówka.Bibliogr. s. 133-135.If T or T* is an algebraically k-quasiclass A operator acting on an infinite dimensional separable Hilbert space and F is an operator commuting with T, and there exists a positive integer n such that Fn has a finite rank, then we prove that Weyl's theorem holds for ∫ (T)+F for every ∫[Formula] H(σ (T)), where H(σ (T)) denotes the set of all analytic functions in a neighborhood of σ (T). Moreover, if T* is an algebraically k-quasiclass A operator, then α-Weyl's theorem holds for ∫(T). Also, we prove that if T or T* is an algebraically k-quasiclass A operator then both the Weyl spectrum and the approximate point spectrum of T obey the spectral mapping theorem for every ∫[Formula] H(σ (T)).Dostępny również w formie drukowanej.KEYWORDS: algebraically k-quasiclass A operator, Weyl's theorem, alpha-Weyl's theorem

ON k -QUASI-PARANORMAL OPERATORS

Abstract. For a positive integer k , an operator T ∈ B(H T k+2 x T k x for all x ∈ H , which is a common generalization of paranormal and quasi-paranormal. In this paper, firstly we prove some inequalities of this class of operators; secondly we give a necessary and sufficient condition for T to be k -quasi-paranormal. Using these results, we prove that: (1) if T n+1 = T n+1 for some positive integer n k , then a k -quasi-paranormal operator T is normaloid; (2) if E is the Riesz idempotent for an isolated point λ 0 of the spectrum of a k -quasi-paranormal operator T , then (i) if λ 0 = 0 , then EH = ker(T − λ 0 ) ; (ii) if λ 0 = 0 , then EH = ker(T k+1 )

On k-Quasiclass A Operators

An operator T∈B(ℋ) is called k-quasiclass A if T∗k(|T2|−|T|2)Tk≥0 for a positive integer k, which is a common generalization of quasiclass A. In this paper, firstly we prove some inequalities of this class of operators; secondly we prove that if T is a k-quasiclass A operator, then T is isoloid and T−λ has finite ascent for all complex number λ; at last we consider the tensor product for k-quasiclass A operators

Weyl's theorem for algebraically k-quasiclass A operators

If $T$ or $T^*$ is an algebraically $k$-quasiclass $A$ operator acting on an infinite dimensional separable Hilbert space and $F$ is an operator commuting with $T$, and there exists a positive integer $n$ such that $F^n$ has a finite rank, then we prove that Weyl's theorem holds for $f(T)+F$ for every $f \in H(\sigma(T))$, where $H(\sigma(T))$ denotes the set of all analytic functions in a neighborhood of $\sigma(T)$. Moreover, if $T^*$ is an algebraically $k$-quasiclass $A$ operator, then $\alpha$-Weyl's theorem holds for $f(T)$. Also, we prove that if $T$ or $T^*$ is an algebraically $k$-quasiclass $A$ operator then both the Weyl spectrum and the approximate point spectrum of $T$ obey the spectral mapping theorem for every $f \in H(\sigma(T))$

On <inline-formula> <graphic file="1029-242X-2009-921634-i1.gif"/></inline-formula>-Quasiclass A Operators

An operator is called -quasiclass if for a positive integer , which is a common generalization of quasiclass . In this paper, firstly we prove some inequalities of this class of operators; secondly we prove that if is a -quasiclass operator, then is isoloid and has finite ascent for all complex number at last we consider the tensor product for -quasiclass operators.</p