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Banach-Lamperti for Kurzweil-Henstock

Thierry De Pauw

TL;DR

This work provides a Banach-Lamperti-type characterization for isometries of the Kurzweil-Henstock function space by identifying surjective linear isometries with bi-$AC$ changes of variable. The authors develop a KH-change-of-variable theorem, embed the problem in an indefinite-integral and Banach-Stone framework, and show that surjective isometries have the explicit form $\mathbf{T}(\pmb{f}) = \pmb{[} \sigma \cdot (f \circ \phi) \cdot \phi' \pmb{]}$ with $\sigma \in \{-1,1\}$ and $\phi$ an increasing bi-$AC$ homeomorphism. They further illuminate the contrasts with Banach-Lamperti for $L_1$ spaces by working in the KH setting, where $\mathbf{KH}(I)$ is not Banach but barrelled, and provide a precise, invariant operator-theoretic description of KH isometries. The results establish a concrete variable-change mechanism for KH-integrable functions, with implications for the structure of KH-operator theory and related functional-analytic methods."

Abstract

We identify isometric isomorphisms of the space of Kurzweil-Henstock integrable functions as bi-absolutely-continuous changes of variable.

Banach-Lamperti for Kurzweil-Henstock

TL;DR

This work provides a Banach-Lamperti-type characterization for isometries of the Kurzweil-Henstock function space by identifying surjective linear isometries with bi- changes of variable. The authors develop a KH-change-of-variable theorem, embed the problem in an indefinite-integral and Banach-Stone framework, and show that surjective isometries have the explicit form with and an increasing bi- homeomorphism. They further illuminate the contrasts with Banach-Lamperti for spaces by working in the KH setting, where is not Banach but barrelled, and provide a precise, invariant operator-theoretic description of KH isometries. The results establish a concrete variable-change mechanism for KH-integrable functions, with implications for the structure of KH-operator theory and related functional-analytic methods."

Abstract

We identify isometric isomorphisms of the space of Kurzweil-Henstock integrable functions as bi-absolutely-continuous changes of variable.
Paper Structure (4 sections, 8 theorems, 42 equations)

This paper contains 4 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Change of variable
  4. Isometries

Key Result

Theorem 1

Let $I$ and $\tilde{I}$ be two cells in $\mathbb{R}$ and $\mathbf{T} : \mathbf{K\!H}(\tilde{I}) \to \mathbf{K\!H}(I)$ a linear operator. The following are equivalent.

Theorems & Definitions (17)

  • Theorem
  • Theorem 2.7
  • proof : Proof that $(A) \Rightarrow (B)$
  • Remark 2.8
  • proof : Proof that $(B) \Rightarrow (A)$
  • Theorem 2.10: A covering theorem
  • Theorem 2.12
  • proof
  • Remark 3.1
  • Theorem 3.3
  • ...and 7 more