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

An operator T 2 B(H) is said to be k-quasi-paranormal operator if kTk+1xk2 · kTk+2xkkTkxk for every x 2 H, k is a natu- ral 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 op- erators which is related to the paranormal operators, we show that every algebraically k-quasi-paranormal operator has Bishop s prop- erty (¯), 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 2 H(¾(T)); (ii) generalized a - Browder s theorem holds for f(S) for every S Á T and f 2 H(¾(S)); (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.