Structure of $n$-quasi left $m$-invertible and related classes of operators

Abstract

Given Hilbert space operators $T, S\in\B$, let $△$ and $δ\in B(\B)$ denote the elementary operators ${△}_{T,S}(X)=(L_T{R}_S-I)(X)=TXS-X$ and ${\delta }_{T,S}(X)=(L_T-R_S)(X)=TX-XS$. Let $d=△$ or $\delta$. Assuming $T$ commutes with $S^{\ast }$, and choosing $X$ to be the positive operator $S^{\ast n}{S}^n$ for some positive integer $n$, this paper exploits properties of elementary operators to study the structure of $n$-quasi $[m,d]$-operators $d_{T,S}^m(X)=0$ to bring together, and improve upon, extant results for a number of classes of operators, amongst them $n$-quasi left $m$-invertible operators, $n$-quasi $m$-isometric operators, $n$-quasi $m$-selfadjoint operators and $n$-quasi $(m,C)$ symmetric operators (for some conjugation $C$ of $\H$). It is proved that $S^n$ is the perturbation by a nilpotent of the direct sum of an operator $S_1^n=(S{|}_{\overline{S^n(\stackrel{˝}{)}}}{)}^n$ satisfying $d_{T_1,S_1}^m(I_1)=0$, $T_1=T{|}_{\overline{S^n(\stackrel{˝}{)}}}$, with the $0$ operator; if also $S$ is left invertible, then $S^n$ is similar to an operator $B$ such that $d_{B^{\ast },B}^m(I)=0$. For power bounded $S$ and $T$ such that $S{T}^{\ast }-T^{\ast }S=0$ and ${△}_{T,S}(S^{\ast n}{S}^n)=0$, $S$ is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators $T,S$ satisfying $d_{T,S}^m(I)=0$, given certain commutativity properties, transfers to operators satisfying $S^{\ast n}{d}_{T,S}^m(I)S^n=0$.

25. arXiv admin note: substantial text overlap with arXiv:1812.00221