The Wold-Type Decomposition for m-Isometries

Abstract

AbstractThe aim of this paper is to study the Wold-type decomposition in the class of m-isometries. One of our main results establishes an equivalent condition for an analytic m-isometry to admit the Wold-type decomposition for $$m\ge 2$$ m ≥ 2 . In particular, we introduce the k-kernel condition which we use to characterize analytic m-isometric operators which are unitarily equivalent to unilateral operator valued weighted shifts for $$m\ge 2$$ m ≥ 2 . As a result, we also show that m-isometric composition operators on directed graphs with one circuit containing only one element are not unitarily equivalent to unilateral weighted shifts. We also provide a characterization of m-isometric unilateral operator valued weighted shifts with positive and commuting weights.