m-isometric weighted shifts with operator weights

The aim of this paper is to study $m$-isometric weighted shifts with operator weights (both unilateral and bilateral). We obtain a characterization of such shifts by polynomials with operator coefficients. The procedure of construction of all $m$-isometric weighted shifts with positive weights is given. We answer the question when the weights of $m$-isometric shifts are commuting. The completion problem for $m$-isometric weighted shifts with operator weights is solved. We characterize $m$-isometric bilateral shifts the adjoints of which are also $m$-isometric. Several relevant examples are provided.