On the Orbit of an m-Isometry

Abstract.

A bounded linear operator T on a Hilbert space H is called an m-isometry for a positive integer m if $\sum\nolimits_{{k = 0}}^{m} {( - 1)^{{m - k}} } \left( {\begin{array}{*{20}c} m \\ k\\ \end{array} } \right)T^{{*k}} T^{k} = 0$. We prove some properties concerning the behaviour of the orbit of an m-isometry. For example, every orbit of an m-isometry is eventually norm increasing and some m-isometries can not be N-supercyclic, that is, there does not exist an N-dimensional subspace E _N such that the orbit of T at E _N is dense in H.