On n-quasi left m-invertible operators

A Hilbert space operator $S\in\B$ is $n$-quasi left $m$-invertible (resp., left $m$-invertible) by $T\in\B$, $m,n \geq 1$ some integers, if $S^{*n}p(S,T)S^n=0$ (resp., $p(S,T)=0$), where $p(S,T)=\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right)T^jS^j}$. Left $m$-invertible and $n$-quasi left $m$-invertible operators share a number of properties. Thus, if $S$ is $n$-quasi left $m$-invertible, then $S^n$ is the perturbation by a nilpotent of the direct sum of a left $m$-invertible with the $0$ operator. In particular, if $T=S^*$ (so that $S$ is $n$-quasi $m$-isomertric) and $|(S|_{\overline{S^n(\H)}})^n|$ is not the identity operator, then $S^n$ is similar to an $m$-isometry. For a power bounded $n$-quasi left $m$-invertible operator $S$ such that $T$ is (also) power bounded. and $ST^*-T^*S=0$, $S$ 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}, $[m,C]$-isometries \cite{CKL}, and (left invertible) $m$-symmetric \cite{CLM} and $m$-selfadjoint \cite{L} operators.