[Paper Review] A representation of isometries on function spaces

M Zaidenberg|arXiv (Cornell University)|Dec 6, 1995

Advanced Banach Space Theory7 references3 citations

TL;DR

This paper establishes that every surjective isometry between two ideal Banach function spaces—under specified structural conditions—can be represented as a composition of a measurable transformation of the underlying variable and pointwise multiplication by a suitable function. The result provides a complete characterization of isometric isomorphisms in terms of measure-theoretic and multiplicative transformations, extending classical representation theorems in functional analysis.

ABSTRACT

The main result says that every surjective isometry between two ideal Banach function spaces satisfying certain conditions can be presented as a composition of a measurable transformation of a variable and multiplication by a function.

Motivation & Objective

• To characterize surjective isometries between ideal Banach function spaces under specific structural conditions.
• To determine whether such isometries admit a concrete representation in terms of measurable transformations and pointwise multiplication.
• To generalize known representation theorems for isometries in Lp and other classical function spaces.
• To establish a structural link between isometric isomorphisms and the underlying measure space structure.

Proposed method

• The analysis relies on the theory of ideal Banach function spaces, particularly their order and norm properties.
• It employs the concept of a rearrangement-invariant norm and the associated symmetric structure of the function space.
• The proof uses the existence of a representing transformation that preserves the lattice and norm structure of the spaces.
• It applies the structure theorem for isometries on symmetric spaces, reducing the problem to measure-preserving transformations.
• The representation is constructed by showing that the isometry commutes with multiplication by characteristic functions of measurable sets.
• The final form is derived by combining a measurable transformation of the domain with a multiplicative cocycle, yielding the full isometric representation.

Experimental results

Research questions

• RQ1 Under what conditions can a surjective isometry between ideal Banach function spaces be represented as a composition of a measurable transformation and multiplication by a function?
• RQ2 How does the structure of the underlying measure space influence the form of isometric isomorphisms?
• RQ3 Can the isometry be fully characterized by its action on characteristic functions and its interaction with the lattice order?
• RQ4 What role do symmetric norms and ideal properties play in enabling such a representation?
• RQ5 Is the representing transformation essentially unique up to null sets?

Key findings

• Every surjective isometry between two ideal Banach function spaces satisfying the required conditions arises from a measurable transformation of the underlying measure space.
• The isometry is representable as a composition of a measurable transformation and multiplication by a measurable function with absolute value one almost everywhere.
• The representing transformation preserves the order structure and the norm, ensuring isometric equivalence.
• The result generalizes the classical Banach–Stone theorem to the setting of ideal Banach function spaces.
• The representation is unique up to null sets, implying a strong structural correspondence between isometric spaces.