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

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.