Opuscula Math. 32, no. 1 (2012), 125-135
http://dx.doi.org/10.7494/OpMath.2012.32.1.125

Opuscula Mathematica

# 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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• Fugen Gao
• Henan Normal University, College of Mathematics and Information Science, Xinxiang, Henan 453007, China
• Tongji University, Department of Mathematics, Shanghai 200092, China
• Xiaochun Fang
• Tongji University, Department of Mathematics, Shanghai 200092, China
• Revised: 2011-02-27.
• Accepted: 2011-04-08.