Embeddings of Müntz Spaces: Composition Operators

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

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