On Quasi-$M$-Hyponormal Operators

Young Min Han, Ju Hee Son

An operator $T$ is called quasi-$M$-hyponormal if there exists a positive real number $M$ such that $T^*(M^2(T-\lambda)^*(T-\lambda))T\geq T^*(T-\lambda)(T-\lambda)^*T$ for all $\lambda \in C$, which is a generalization of $M$-hyponormality. We consider the local spectral properties for quasi-$M$-hyponormal operators and Weyl type theorems for algebraically quasi-$M$-hyponormal operators, respectively. It is also proved that if $T$ is an algebraically quasi-$M$-hyponormal operator, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.