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

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