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On n-quasi left m-invertible operators

Research paper by B. P. Duggal

Indexed on: 01 Dec '18Published on: 01 Dec '18Published in: arXiv - Mathematics - Functional Analysis



Abstract

A Hilbert space operator $S\in\B$ is $n$-quasi left $m$-invertible by $T\in\B$, for some integers\hspace{3mm} $m,n \geq 1$, \hspace{3mm}if $S^{*n}p(S,T)S^n=0$, \hspace{3mm} where $p(S,T)=$\newline$\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right)T^jS^j}$ (and $p(S,T)=0$ defines the class of operators $S\in\B$ which are left $m$-invertible by $T\in\B$). We prove that $n$-quasi left $m$-invertible operators share a number of properties with left $m$-invertible operators. Thus, for a power bounded $n$-quasi left $m$-invertible operator $S$ such that $T$ is (also) power bounded and $ST^*-T^*S=0$, $S^n$ is similar to the direct sum of a power bounded left $m$-invertible operator (with a power bounded left inverse) and the $0$ operator, hence $S$ is polaroid (i.e., isolated points of the spectrum are poles); the product of an $n$-quasi left $m_1$-invertible operator with a left $m_2$-invertible operator, given certain commutativity properties, is $n$-quasi left $(m_1+m_2-1)$-invertible; again, if $ST^*-T^*S=0$ and $N$ is an $n_1$-nilpotent which commutes with $S$, then $T$ is an $(n+n_1-1)$-quasi left $(m+n_1-1)$-inverse of $S+N_1$. These results have applications to $n$-quasi $m$-isometries \cite{AS} (it is seen that the $n$-th power of an $n$-quasi $m$-isometry is similar to an $m$-isometry), $[m,C]$-isometries \cite{CKL}, and $m$-symmetric \cite{CLM} and $m$-selfadjoint \cite{L} operators.