(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∑k=0m(−1)m−k(mk)T⁎kATk=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.