Boundary multipliers of a family of M\"obius invariant function spaces

TL;DR

This paper characterizes the multipliers and spectra of multiplication operators on a family of M"obius invariant function spaces defined on the unit circle, expanding understanding of their structure and operator behavior.

Contribution

It provides a complete description of multipliers and spectra for the spaces al^p_s(\u00a4) on the unit circle, a new insight into these M"obius invariant spaces.

Findings

Complete characterization of multipliers between al^p_s() spaces.

Determination of the spectra of multiplication operators on these spaces.

Enhanced understanding of the structure and operator theory of M"obius invariant function spaces.

Abstract

For $1<p<\infty$ and $0<s<1$, let $\mathcal{Q}^p_ s (\mathbb{T})$ be the space of those functions $f$ which belong to $ L^p(\mathbb{T})$ and satisfy \[ \sup_{I\subset \mathbb{T}}\frac{1}{|I|^s}\int_I\int_I\frac{|f(\zeta)-f(\eta)|^p}{|\zeta-\eta|^{2-s}}|d\zeta||d\eta|<\infty, \] where $|I|$ is the length of an arc $I$ of the unit circle $\mathbb{T}$ . In this paper, we give a complete description of multipliers between $\mathcal{Q}^p_ s (\mathbb{T})$ spaces. The spectra of multiplication operators on $\mathcal{Q}^p_ s (\mathbb{T})$ are also obtained.