  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".
