Weyl type theorem and spectrum for $(p,k)$-quasiposinormal operators

Let $T$ be a $(p,k)$-quasiposinormal operator on a complex Hilbert space $\mathcal{H}$, i.e $T^{*k}(c^{2}(T^{*} T)^{p}-(T T^{*})^{p})T^{k} \geq 0$ for a positive integer $p \in (0,1]$, some $c > 0$ and a positive integer $k$. In this paper, we prove that the spectral mapping theorem for Weyl spectrum holds for $(p, k)$ - quasiposinormal operators. We show that the Weyl type theorems holds for $(p,k)$- quasiposinormal. We prove that if $T^{*}$ is $(p,k)$-quasiposinormal, then generalized $a$-Weyl's theorem holds for $T$. Also we prove that $\sigma_{jp}(T)-\{0\} = \sigma_{ap}(T)-\{0\}$ holds for $(p,k)$-quasiposinormal operator.