# On n-quasi left m-invertible operators

Research paper by **B. P. Duggal**

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

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#### 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.