Some results on higher orders quasi-isometries

Abstract

The purpose of the present paper is to pursue further study of a class of linear bounded operators, known as $n$-quasi-$m$-isometric operators acting on an infinite complex separable Hilbert space ${\mathcal H}$. We give an equivalent condition for any $T$ to be $n$-quasi-$m$-isometric operator. Using this result we prove that any power of an $n$-quasi-$m$-isometric operator is also an $n$-quasi-$m$-isometric operator. In general the converse is not true. However, we prove that if $T^r$ and $T^{r+1}$ are $n$-quasi-$m$-isometries for a positive integer $r$, then T is an $n$-quasi-$m$-isometric operator. We study the sum of an $n$-quasi-$m$-isometric operator with a nilpotent operator. We also study the product and tensor product of two $n$-quasi-$m$-isometries. Further, we define $n$-quasi strict $m$-isometric operators and prove their basic properties.