Spectrum of Quasi-Class (A,k) Operators

Spectrum of Quasi-Class (A,k) Operators

An operator T∈B(ℋ) is called quasi-class (A,k) if T∗k(|T2|−|T|2)Tk≥0 for a positive integer k, which is a common generalization of class A. In this paper, firstly we consider some spectral properties of quasi-class (A,k) operators; it is shown that if T is a quasi-class (A,k) operator, then the nonz...

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Veröffentlicht in:ISRN mathematical analysis 2011-01, Vol.2011 (2011), p.1-10
Hauptverfasser: Li, Xiaochun, Gao, Fugen, Fang, Xiaochun
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Hilbert space
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description An operator T∈B(ℋ) is called quasi-class (A,k) if T∗k(|T2|−|T|2)Tk≥0 for a positive integer k, which is a common generalization of class A. In this paper, firstly we consider some spectral properties of quasi-class (A,k) operators; it is shown that if T is a quasi-class (A,k) operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigenspaces corresponding to distinct eigenvalues of T are mutually orthogonal, and the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. Secondly, we show that Putnam's theorems hold for class A operators. Particularly, we show that if T is a class A operator and either σ(|T|) or σ(|T∗|) is not connected, then T has a nontrivial invariant subspace.
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title Spectrum of Quasi-Class (A,k) Operators
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