GENERALIZED WEYL'S THEOREM FOR ALGEBRAICALLY <TEX>$k$</TEX>-QUASI-PARANORMAL OPERATORS

Abstract

An operator <TEX>$T\;{\varepsilon}\;B(\mathcal{H})$</TEX> is said to be <TEX>$k$</TEX>-quasi-paranormal operator if <TEX>$||T^{k+1}x||^2\;{\leq}\;||T^{k+2}x||\;||T^kx||$</TEX> for every <TEX>$x\;{\epsilon}\;\mathcal{H}$</TEX>, <TEX>$k$</TEX> 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 <TEX>$k$</TEX>-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically <TEX>$k$</TEX>-quasi-paranormal operator has Bishop's property (<TEX>$\beta$</TEX>), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for <TEX>$f(T)$</TEX> for every <TEX>$f\;{\epsilon}\;H({\sigma}(T))$</TEX>; (ii) generalized a - Browder's theorem holds for <TEX>$f(S)$</TEX> for every <TEX>$S\;{\prec}\;T$</TEX> and <TEX>$f\;{\epsilon}\;H({\sigma}(S))$</TEX>; (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.