Some results on higher orders quasi-isometries

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 H. space 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-isometrics 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-isometrics. Further, we define n-quasi strict m-isometric operators and prove their basic properties.