An isometry plus a nilpotent operator is an -isometry. Applications

We prove that if an isometry A and a nilpotent operator Q of order n commute, then A + Q is a strict ( 2 n − 1 ) -isometry. As an application of the main result, we prove that A + Q cannot be N -supercyclic for any N . Also, we find an m -isometry with prescribed spectrum K , where K is the closed unit disk or a compact subset of the unit circle.