Communications of the
Korean Mathematical Society CKMS

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