The Wold-Type Decomposition for m-Isometries

The 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≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 2$$\end{document}. 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≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 2$$\end{document}. 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.