Embeddings of M\"{u}ntz Spaces: Composition Operators

TL;DR

This paper investigates the properties of embeddings of Müntz spaces into L^2 spaces with respect to measures and explores how these properties influence the behavior of composition operators, including criteria for boundedness, compactness, and Schatten class membership.

Contribution

It provides new criteria for the boundedness, compactness, and Schatten class membership of composition operators on Müntz spaces based on embedding properties.

Findings

Criteria for bounded composition operators on Müntz spaces.

Conditions for compactness of composition operators.

Characterization of Schatten--von Neumann class membership.

Abstract

Given a strictly increasing sequence $\Lambda=(\lambda_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty$, the M\"untz spaces $M_\Lambda^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{\lambda_n}$. We discuss how properties of the embedding $M_\Lambda^2\subset L^2(\mu)$, where $\mu$ is a finite positive Borel measure on the interval $[0,1]$, have immediate consequences for composition operators on $M^2_\Lambda$. We give criteria for composition operators to be bounded, compact, or to belong to the Schatten--von Neumann ideals.