Exponentially m-isometric operators on Hilbert spaces

Abstract

For a positive integer m, a bounded linear operator T on a Hilbert space is called an exponentially m-isometric operator if ∑k=0m(−1)m−k(mk)ekT⁎ekT=0. For 1≤n≤m, skew-n-selfadjoint operators, nilpotent operators of order less than or equal to [m+12], the greatest integer not greater than m+12, and 2πi multiples of idempotents are main examples of such operators. We establish a decomposition theorem for strict exponentially m-isometric operators with finite spectrum and prove that they are exponentially isometric m-Jordan. Finally, the dynamics of this operator will be considered. We will show that there is no N-supercyclic exponentially m-isometric operator on an infinite-dimensional Hilbert space.