Generalized Weyl’s theorem for some classes of operators

Abstract. Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set σBw(A) of all λ ∈ C such that A−λI is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl’s theorem σBw(A) = σ(A) \ E(A), and the B-Weyl spectrum σBw(A) of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl’s theorem holds for hermitian op-erators. Weyl’s theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl’s theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee’s results to al-gebraically paranormal operators. In [19] the authors showed that Weyl’s theorem holds for algebraically p-hyponormal operators. As Berkani has shown in [5], if the generalized Weyl’s theorem holds for A, then so does Weyl’s theorem. In this paper all the above re-sults are generalized by proving that generalized Weyl’s theorem holds for the case where A is an algebraically (p, k)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators. 1