Weyl's Theorem for Algebraically (𝑝, 𝑘)-Quasihyponormal Operators
Abstract Let 𝐴 be a bounded linear operator acting on a Hilbert space 𝐻. The 𝐵-Weyl spectrum of 𝐴 is the set σ 𝐵𝑤(𝐴) of all ⋋ ∈ ℂ such that 𝐴 – ⋋𝐼 is not a 𝐵-Fredholm operator of index 0. Let 𝐸(𝐴) be the set of all isolated eigenvalues of 𝐴. Recently, in [Berkani and Arroud, J. Aust. Math. Soc. 76: 291–302, 2004] the author showed that if 𝐴 is hyponormal, then 𝐴 satisfies the generalized Weyl's theorem σ 𝐵𝑤(𝐴) = σ(𝐴) \ 𝐸(𝐴), and the 𝐵-Weyl spectrum σ 𝐵𝑤(𝐴) of 𝐴 satisfies the spectral mapping theorem. Lee [Han, Proc. Amer. Math. Soc. 128: 2291–2296, 2000] showed that Weyl's theorem holds for algebraically hyponormal operators. In this paper the above results are generalized to an algebraically (𝑝, 𝑘)-quasihyponormal operator which includes an algebraically hyponormal operator.