AboundedlinearoperatorTonaHilbertspaceHiscalledan(A,m)-isometry, for somepositiveoperatorAonHandintegermif m k=0 (−1)m−k m k T∗kATk=0. We give someproperties of (A,m)-isometries. Inparticular, we focus on spectral properties andthe relationbetween(A,m)-isometriesandm-isometries. Also,weobtainsomedynamic propertiesof (A,m)-isometriesas: anegativeanswerto[22,Question1]withanexampleofan A-isometricwhich isN-supercyclicandsufficient conditions for an(A,m)-isometrytobenot N-supercyclic. Moreover,weprovethat theperturbationof (A,m)-isometrybyabiggerclass thannilpotentoperatorsisnotN-supercyclic.
