Weyl's theorem for algebraically k-quasiclass A operators

Fugen Gao
Xiaochun Fang

Abstract. 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 \(f \in H(\sigma(T))\).

Keywords: algebraically \(k\)-quasiclass \(A\) operator, Weyl's theorem, \(\alpha\)-Weyl's theorem.

Mathematics Subject Classification: 47B20.

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