On the Wiener-Pitt phenomenon for algebra of Rajchman multipliers on Hardy space

TL;DR

This paper investigates the spectral properties of Rajchman measures within the multiplier algebra of Hardy space, revealing non-natural spectra for measures with Minkowski dimension zero.

Contribution

It demonstrates that positive Rajchman measures of Minkowski dimension zero have non-natural spectra in the multiplier algebra of Hardy space, providing new insights into spectral theory and harmonic analysis.

Findings

Positive Rajchman measures of Minkowski dimension zero have non-natural spectra.

The proof involves estimating the norm of convolution operators on Hardy space.

The results connect measure theory with spectral properties in harmonic analysis.

Abstract

We show that any positive Rajchman measure of Minkowski dimension $0$ has a non-natural spectrum as an element of the multiplier algebra of $H^{1}_{0}(\T)$. The proof is based on the estimation of the norm of the convolution operator given by a singular measure on $H_{0}^{1}(\T)$.