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Compactness and Spectral Properties of Multiplier Operators in the Walsh System

Michael Ruzhansky, Sergo A. Episkoposian, Rafik Yeghoyan

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.

Compactness and Spectral Properties of Multiplier Operators in the Walsh System

Abstract

We investigate compactness and spectral properties of multiplier operators associated with the Walsh system in the spaces , . Building upon previously established criteria for boundedness of Walsh multipliers, we prove an exact compactness criterion in the regime for all (assuming boundedness of the multiplier), and also in the regime for . The key result states that compactness is equivalent to the condition for the multiplier symbol. We also examine in detail the point spectrum and derive strict spectral inclusions; in the Hilbert space case we obtain a complete description of the spectrum. For , we emphasize the limitations of transferring "diagonal" arguments and formulate results in a form that does not admit incorrect generalizations.
Paper Structure (20 sections, 10 theorems, 65 equations)

This paper contains 20 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Walsh system and Walsh–Fourier coefficients
  4. Multiplier operators
  5. Analysis and synthesis inequalities (Walsh)
  6. Bounded symbols and boundedness of multipliers
  7. Norm of the difference of Walsh functions
  8. Compactness of multiplier operators
  9. Compactness criterion in the $L^p\to L^p$ regime for $1<p<2$
  10. Compactness in the $L^p\to L^2$ regime for $2<p<\infty$
  11. Spectral properties
  12. Point spectrum
  13. Spectrum in the case $p=2$
  14. Spectral inclusions for $p\neq 2$
  15. Compact multipliers and spectrum
  16. ...and 5 more sections

Key Result

Lemma 2.3

Let $1<p\le 2$ and $p'=\frac{p}{p-1}$. Then there exists a constant $C_{HY}(p)>0$ such that for all $f\in L^p[0,1]$

Theorems & Definitions (30)

  • Definition 2.1: Rademacher and Walsh--Paley functions
  • Definition 2.2: Walsh multiplier
  • Lemma 2.3: Hausdorff–Young for Walsh
  • proof
  • Lemma 2.4: Synthesis in $L^p[0,1]$ for Walsh
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 20 more