# Isometric Operators on Variable-Exponent Discrete Lebesgue Spaces

## TL;DR

This paper characterizes isometric operators on variable-exponent discrete Lebesgue spaces, revealing both similarities and differences with fixed-exponent spaces, and introduces a novel class of isometries determined by set-mappings and functions.

## Contribution

It introduces a new class of isometries on variable-exponent spaces and characterizes their structure, extending known results from fixed-exponent spaces.

## Key findings

- New class of isometries characterized by set-mappings and functions
- Shift operators are isometric only under restrictive conditions
- Structural similarities and differences with fixed-exponent spaces highlighted

## Abstract

We investigate the structure of norm-preserving and linear but not necessarily surjective operators on variable-exponent, discrete Lebesgue spaces. A certain class of isometries, novel to this work, are especially considered; this class completely coincides with all isometries when the Lebesgue space is classical, i.e. of a fixed-exponent. For said isometries it is shown that their actions are completely determined by pairs consisting of set-mappings and bounded functions on $\mathbb{N}$. This result recovers the previously-known structure of isometries on fixed-exponent spaces as a special case. In the second part, we show that another wide class of operators, including shift operators, are only isometric under very restrictive conditions on the exponent sequence. Together these results serve to highlight the striking similarities and yet radical differences between isometric operators on fixed- and variable-exponent spaces.