Linear preservers of majorization on $\ell^p(I)$

TL;DR

This paper extends the concept of majorization to infinite-dimensional $ ext{ell}^p$ spaces using doubly stochastic operators and characterizes all bounded linear maps that preserve majorization for $p$ in (1, +∞).

Contribution

It introduces a new extension of majorization to infinite-dimensional spaces and characterizes the structure of all majorization-preserving bounded linear maps for certain p-values.

Findings

Majorization extended to $ ext{ell}^p(I)$ using doubly stochastic operators.

Complete characterization of majorization-preserving bounded linear maps for $p eq 1, ext{or} eq ext{infinity}$.

Provides a framework for understanding linear preservers in infinite-dimensional spaces.

Abstract

In this paper, using doubly stochastic operators, we have extended the notion of majorization to the space $\ell^p(I)$, where $I$ is assumed to be an infinite set, and then, in the case $p\in (1,+\infty)$, characterize the structure of all bounded linear maps on this space which preserves majorization.