On power Drazin normal and Drazin quasi-normal Hilbert space operators

Abstract

A Drazin invertible Hilbert space operator $T\in \B$, with Drazin inverse $T_d$, is $(n,m)$-power D-normal, $T\in [(n,m) DN]$, if $[T_d^n,T^{*m}]=T^n_dT^{*m}-T^{*m}T_d^n=0$; $T$ is $(n,m)$-power D-quasinormal, $T\in [(n,m) DQN]$, if $[T_d^n,T^{*m}T]=0$. Operators $T\in [(n,m) DN]$ have a representation $T=T_1\oplus T_0$, where $T_1$ is similar to an invertible normal operator and $T_0$ is nilpotent. Using this representation, we have a keener look at the structure of $[(n,m) DN]$ and $[(n,m) DQN]$ operators. It is seen that $T\in [(n,m) DN]$ if and only if $T\in [(n,m) DQN]$, and if $[T,X]=0$ for some operators $X\in\B$ and $T\in [(1,1) DN]$, then $[T^*_d,X]=0$. Given simply polar operators $S, T\in [(1,1) DN]$ and an operator $A=\left(\begin{array}{clcr} T&C 0&S \end{array}\right) \in B(\H\oplus\H)$, $A\in [(1,1) DN]$ if and only if $C$ has a representation $C=0\oplus C_{22}$.