Embeddings of Müntz 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 Müntz 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 $M_{\Lambda }^2$. We give criteria for composition operators to be bounded, compact, or to belong to the Schatten--von Neumann ideals.

arXiv comment15 Pages; Integral Equations Operator Theory, Springer, 2012