(A, m)-isometries on Hilbert spaces

A bounded linear operator T on a Hilbert space H is called an (A, m)-isometry, for some positive operator A on H and integer m if Sigma(m)(k=0)(-1)(m-k)((m)(k))T*(k) AT(k) = 0 We give some properties of (A, m)-isometries. In particular, we focus on spectral properties and the relation between (A, m')-isometries and m-isometries. Also, we obtain some dynamic properties of (A, m)-isometries as: a negative answer to [22, Question 1] with an example of an A-isometric which is N-supercyclic and sufficient conditions for an (A, m)-isometry to be not N-supercyclic. Moreover, we prove that the perturbation of (A, m) -isometry by a bigger class than nilpotent operators is not N-supercyclic. (C) 2017 Elsevier Inc. All rights reserved.