Compactness and Spectral Properties of Multiplier Operators in the Walsh System

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.