$n$-Multiplicity and spectral properties for $(M, k)$-quasi-$*$-class $Q$ operators

In the present article, we introduce a new class of operators which will be called the class of $(M, k)$-quasi-$*$-class $Q$ operators. An operator $A\in B(H)$ is said to be $(M, k)$-quasi-$*$-class $Q$ for certain integer $k$, if there exists $M>0$ such that
$$ A^{*k}(MA^{*2}A^2-2AA^*+I)A^k\geqslant0. $$
Some properties of this class of operators are shown. It is proved that the considered class contains the class of $k$-quasi-$*$-class $\mathbb{A}$ operators. The decomposition of such operators, their restrictions on invariant subspaces, the $n$-multicyclicity and some spectral properties are also presented. We also show that if $\lambda\in\mathbb{C}$, $\lambda\ne0$ is an isolated point of the spectrum of $A$, then the Riesz idempotent $E$ for $\lambda$ is self-adjoint, and verifies $EH=ker(A-\lambda)=ker(A-\lambda)^*$.