n-Multiplicity and Spectral Properties for (M,k)-Quasi-*-Class Q Operators

Abstract

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

\(A^k(MA^*2A^2 - 2AA^* + I)A^k \geq 0\).

Some properties of this class of operators are shown. It is proved that the considered class contains the class of \(k\)-quasi-\(*\)-class \(A\) 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 \(\lambda \in \mathbb{C}, \lambda \neq 0\) is an isolated point of the spectrum of \(A\), then the Riesz idempotent \(E\) for \(\lambda\) is self-adjoint, and verifies \(EH = \ker(A - \lambda) = \ker(A - \lambda)^*\).