m-isometric operators and their local properties

Abstract In this paper we give necessary and sufficient conditions for a bounded linear Hilbert space operator to be an m-isometry for an unspecified m written in terms of conditions that are applied to “one vector at a time”. We provide criteria for orthogonality of generalized eigenvectors of an (a priori unbounded) linear operator T on an inner product space that correspond to distinct eigenvalues of modulus 1. We also discuss a similar question of when Jordan blocks of T corresponding to distinct eigenvalues of modulus 1 are “orthogonal”.