GENERALIZED WEYL'S THEOREM FOR ALGEBRAICALLY k-QUASI-PARANORMAL OPERATORS

Abstract

An operator T ∈ B(H) is said to be k-quasi-paranormal operator if (cid:107)T k+1x(cid:107)2 ≤ (cid:107)T k+2x(cid:107)(cid:107)T kx(cid:107) for every x ∈ H, k is a natural number. This class of operators contains the class of paranormal operators and the class of quasi class A operators. In this paper, using the operator matrix representation of k-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically k-quasi-paranormal operator has Bishop's property (β), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for f (T ) for every f ∈ H(σ(T )); (ii) generalized a Browder's theorem holds for f (S) for every S ≺ T and f ∈ H(σ(S)); (iii) the spectral mapping theorem holds for the B Weyl spectrum of T .

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