Compactness and Spectral Properties of Multiplier Operators in the Walsh System

TL;DR

This paper characterizes when Walsh multiplier operators are compact on $L^p$ spaces, linking compactness to the decay of the multiplier symbol, and fully describes their spectral properties, especially in the Hilbert space case.

Contribution

It provides an exact compactness criterion for Walsh multipliers in $L^p$ spaces and analyzes their spectral properties, including a complete spectrum description for $p=2$.

Findings

Compactness is equivalent to the multiplier symbol tending to zero.

Complete spectral description for the case $p=2$.

Limitations of diagonal argument transfer for $p eq 2$.

Abstract

We investigate compactness and spectral properties of multiplier operators associated with the Walsh system in the spaces $L^p[0,1]$, $1<p<\infty$. Building upon previously established criteria for boundedness of Walsh multipliers, we prove an exact compactness criterion in the $L^p\to L^p$ regime for all $1<p<\infty$(assuming boundedness of the multiplier), and also in the $L^p\to L^2$ regime for $2<p<\infty$. The key result states that compactness is equivalent to the condition $a_n\to 0$ for the multiplier symbol. We also examine in detail the point spectrum and derive strict spectral inclusions; in the Hilbert space case $p=2$ we obtain a complete description of the spectrum. For $p\neq 2$, we emphasize the limitations of transferring "diagonal" arguments and formulate results in a form that does not admit incorrect generalizations.