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Revisiting products and powers of $(m,p)$ and $(m,\infty)$-isometries

Michael Mackey

TL;DR

This work revisits powers and products of $(m,p)$-isometries, and extends the analysis to the $(m,\infty)$ case by translating operator properties into polynomial- and combinatorics-based frameworks. It provides elementary, polynomial-interpolation proofs showing that powers of an $(m,p)$-isometry remain in the same class and that the product of commuting $(m,p)$- and $(n,p)$-isometries is a $(m+n-1,p)$-isometry, while the $p=\infty$ setting is governed by binary-sequence dynamics rather than straightforward operator theory. A key methodological contribution is the two-variable polynomial-matrix approach, which simultaneously handles row/column interpolation and enables the product theorem. The paper also develops a binary-sequence/graph representation for $(m,\infty)$-isometries, establishes periodic and nonperiodic patterns (e.g., $(1100)$ for $(3,\infty)$), counts realizable sequences via Fibonacci-type relations, and outlines constructions of $(m,\infty)$-operators that realize targeted patterns, shedding light on the structural depth of these isometries.

Abstract

We review known results concerning powers and products of $(m,p)$-isometries with a view to providing elementary proofs based on properties of polynomials. We consider also the situation when $p=\infty$ where we find elements of graph theory and combinatorics arise naturally.

Revisiting products and powers of $(m,p)$ and $(m,\infty)$-isometries

TL;DR

This work revisits powers and products of -isometries, and extends the analysis to the case by translating operator properties into polynomial- and combinatorics-based frameworks. It provides elementary, polynomial-interpolation proofs showing that powers of an -isometry remain in the same class and that the product of commuting - and -isometries is a -isometry, while the setting is governed by binary-sequence dynamics rather than straightforward operator theory. A key methodological contribution is the two-variable polynomial-matrix approach, which simultaneously handles row/column interpolation and enables the product theorem. The paper also develops a binary-sequence/graph representation for -isometries, establishes periodic and nonperiodic patterns (e.g., for ), counts realizable sequences via Fibonacci-type relations, and outlines constructions of -operators that realize targeted patterns, shedding light on the structural depth of these isometries.

Abstract

We review known results concerning powers and products of -isometries with a view to providing elementary proofs based on properties of polynomials. We consider also the situation when where we find elements of graph theory and combinatorics arise naturally.

Paper Structure

This paper contains 7 sections, 13 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Powers of $(m,p)$-isometries
  3. Polynomial matrices
  4. Products of $(m,p)$-isometries
  5. What about $(m,\infty)$-isometries?
  6. $(3,\infty)$ binary sequences
  7. $(4,\infty)$ binary sequences

Key Result

Theorem 1

An operator $T:X\to X$ is an $(m, p)$-isometry if and only if, for every $x\in X$ there is a polynomial $q_x\in \mathcal{P}^{m-1}$ such that $\lVert T^n x \rVert^p=q_x(n)$ for all $n\in\mathbb{N}^0$.

Figures (3)

  • Figure 1: Directed graph for $(3,\infty)$ sequences
  • Figure 2: Graph for $(4,\infty)$ sequence
  • Figure 3: Graph for $(4,\infty)$ sequence, degree 2 vertices removed

Theorems & Definitions (19)

  • Theorem 1: MR2852193
  • Theorem 2: MR2911496
  • Proposition 3: MR2859754
  • Lemma 4
  • Theorem 5
  • Remark 6
  • Definition 7
  • Example 8
  • Proposition 9
  • Corollary 10
  • ...and 9 more