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 has a finite rank, then we prove that Weyl’s theorem holds for f(T )+F for every f ∈ 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 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 ∈ H(σ(T )).