m-isometric weighted shifts with operator weights

TL;DR

The paper addresses the characterization and construction of $m$-isometric weighted shifts with operator weights for both unilateral and bilateral cases. It develops a polynomial-with-operator-coefficient framework, proving that $m$-isometry is equivalent to the existence of a polynomial $p\in \mathbf{B}(H)_{m-1}[z]$ with $p(0)=I$ and $p(n)=|S_n\cdots S_1|^2$, and it provides a concrete construction method for all such shifts with positive invertible weights. It analyzes weight commutativity, solves the completion problem for preset initial weights, and extends results to bilateral shifts, showing that the adjoint being $m$-isometric hinges on the invertibility of the same operator polynomial, which yields a negative answer to a prior conjecture for scalar weights. Together, these results give explicit criteria, norm formulas, and examples that deepen the theory of operator-valued weighted shifts and $m$-isometries, with implications for structure and classification in operator theory.

Abstract

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.

Theorems & Definitions (36)

• Lemma 2.1
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['ThmCharOfMIsometricUWSByOperatorPolynomial']}
• Corollary 3.3
• proof
• Theorem 3.4
• proof