Geodesic
$n$-Multiplicity and spectral properties for $(M, k)$-quasi-$*$-class $Q$ operators
Eurasian mathematical journal, Tome 14 (2023) no. 2, pp. 79-93.

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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 $EH=ker(A-\lambda)=ker(A-\lambda)^*$. 
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A. Nasli Bakir; S. Mecheri. $n$-Multiplicity and spectral properties for $(M, k)$-quasi-$*$-class $Q$ operators. Eurasian mathematical journal, Tome 14 (2023) no. 2, pp. 79-93. http://geodesic.mathdoc.fr/item/EMJ_2023_14_2_a4/

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