Spectral and topological properties of linear operators on a Hilbert space

We introduce the class of $(M, k)$-quasi-$*$-paranormal operators on a Hilbert space $H$. This extends classes $*$-paranormal and $k$-quasi-$*$-paranormal operators. An operator $T$ complex is called if there exists $M>0$ such that \begin{equation*} \sqrt{M}\left\Vert T^{k+2}x\right\Vert \left\Vert T^{k}x\right\Vert \geq T^{\ast }T^{k}x\right\Vert ^{2} \end{equation*} for all $x\in H.$ In present article, we give matrix representation operator. The compactness, invariant subspace, some topological properties this are studied. Several also presented.