Semigroups of composition operators and integral operators on mixed norm spaces

TL;DR

This paper characterizes strongly continuous semigroups of composition operators on mixed norm spaces, exploring their behavior on different subspaces and introducing integral operators to understand their properties.

Contribution

It provides a comprehensive analysis of semigroups on mixed norm spaces, introducing integral operators and identifying the maximal subspace for strong continuity.

Findings

Semigroups are strongly continuous on certain subspaces of $H(p,q,eta)$.

Integral operators are characterized in terms of boundedness and compactness.

The maximal space for strong continuity is either $H(p,0,eta)$ or non-separable.

Abstract

We characterize the semigroups of composition operators that are strongly continuous on the mixed norm spaces $H(p,q,\alpha)$. First, we study the separable spaces $H(p,q,\alpha)$ with $q<\infty,$ that behave as the Hardy and Bergman spaces. In the second part we deal with the spaces $H(p,\infty,\alpha),$ where polynomials are not dense. To undertake this study, we introduce the integral operators, characterize its boundedness and compactness, and use its properties to find the maximal closed linear subspace of $H(p,\infty,\alpha)$ in which the semigroups are strongly continuous. In particular, we obtain that this maximal space is either $H(p,0,\alpha)$ or non-separable, being this result the deepest one in the paper.