A Note on an elementary Operator with Quasi-M-Hyponormal Operator Entries

A bounded operator T on the Hilbert space H is called quasi-M-hyponormal if there exists a positive real number M such that $T^{*}\left(M^{2}(T-\lambda)^{*}(T-\lambda)\right) T \geq T^{*}(T-\lambda)(T-\lambda)^{*} T$ for all $\lambda \in \mathbb{C}$. In this paper we prove that if T is quasi-Mhyponormal then T is polaroid and T has Bishop’s property $(\beta)$. We show that if $E_{\mu}$ is the Riesz idempotent for a nonzero isolated point $\mu$ of the spectrum of T, then $E_{\mu}$ is self-adjoint and $E_{\mu}=N(T-\mu)=N(T-\mu)^{*}$. Let $A, B^{*} \in B(H)$ be quasi-M-hyponormal operators, and let $d_{A B} \in B(B(H))$ denote either the generalized derivation $\delta_{A B}(X)=A X-X B$ or the elementary operator $\Delta_{A B}(X)=A X B-X$. We prove that $N\left(d_{A B}-\lambda\right) \subseteq N\left(d_{A B}^{*}-\bar{\lambda}\right)$, for all non null complex $\lambda$. An engineering application is also presented.