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 $[T_d^n,T^{\ast m}]=T_d^n{T}^{\ast m}-T^{\ast m}{T}_d^n=0$; $T$ is $(n,m)$-power D-quasinormal, $T\in [(n,m)DQN]$, if $[T_d^n,T^{\ast m}T]=0$. 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 $[T_d^{\ast },X]=0$. Given simply polar operators $S,T\in [(1,1)DN]$ and an operator $A=\left(\begin{array}{clc}T& C0& S\end{array}\right)\in B(\stackrel{˝}{\oplus }\stackrel{˝}{)}$, $A\in [(1,1)DN]$ if and only if $C$ has a representation $C=0\oplus C_{22}$.

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