On n-quasi- [m,C] $[m,C]$-isometric operators

Abstract

Abstract For positive integers m and n, an operator T∈B(H) $T \in B ( H )$ is said to be an n-quasi- [m,C] $[m,C]$-isometric operator if there exists some conjugation C such that T∗n(∑j=0m(−1)j(mj)CTm−jC.Tm−j)Tn=0 . In this paper, some basic structural properties of n-quasi- [m,C] $[m,C]$-isometric operators are established with the help of operator matrix representation. As an application, we obtain that a power of an n-quasi- [m,C] $[m,C]$-isometric operator is again an n-quasi- [m,C] $[m,C]$-isometric operator. Moreover, we show that the class of n-quasi- [m,C] $[m,C]$-isometric operators is norm closed. Finally, we examine the stability of n-quasi- [m,C] $[m,C]$-isometric operator under perturbation by nilpotent operators commuting with T.

Keywords

• n-quasi- [ m , C ] $[m,C]$ -isometric operator
• Perturbation
• Nilpotent operator