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