Weyl's type theorems for algebraically $(p,k)$-quasihyponormal operators
Commun. Korean Math. Soc. 2012 Vol. 27, No. 1, 77-95
Printed March 1, 2012
Mohammad Hussein Mohammad Rashid and Mohd Salmi Mohd Noorani
Mu'tah University, Universiti Kebangsaan Malaysia
Abstract : For a bounded linear operator $T$ we prove the following assertions: (a) If $T$ is algebraically $(p,k)$-quasihyponormal, then $T$ is $a$-isoloid, polaroid, reguloid and $a$-polaroid. (b) If $T^*$ is algebraically $(p,k)$-quasihyponormal, then $a$-Weyl's theorem holds for $f(T)$ for every $f\in Hol(\sigma(T)),$ where $Hol(\sigma(T))$ is the space of all functions that analytic in an open neighborhoods of $\sigma(T)$ of $T$. (c) If $T^*$ is algebraically $(p,k)$-quasihyponormal, then generalized $a$-Weyl's theorem holds for $f(T)$ for every $f\in Hol(\sigma(T))$. (d) If $T$ is a $(p,k)$-quasihyponormal operator, then the spectral mapping theorem holds for semi-$B$-essential approximate point spectrum $\sigma_{SBF_{+}^{-}}(T),$ and for left Drazin spectrum $\sigma_{lD}(T)$ for every $f\in Hol(\sigma(T)).$
Keywords : $(p,k)$-quasihyponormal, single valued extension property, Fredholm theory, Browder's theory, spectrum
MSC numbers : 47A10, 47A12, 47A55, 47B20
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