Riesz-Kolmogorov type compactness criteria in function spaces with applications

TL;DR

This paper extends the Riesz-Kolmogorov compactness criteria to various function spaces, providing new tools for analyzing compactness and operator properties in complex analysis and functional analysis.

Contribution

It introduces generalized compactness criteria applicable to multiple function spaces, enabling new characterizations of compact operators like Toeplitz and Hankel operators.

Findings

Classified precompact sets in several function spaces.

Characterized compact Toeplitz operators on Bergman space.

Established compactness of Hankel operators on Hardy space.

Abstract

We present forms of the classical Riesz-Kolmogorov theorem for compactness that are applicable in a wide variety of settings. In particular, our theorems apply to classify the precompact subsets of the Lebesgue space $L^2$, Paley-Wiener spaces, weighted Bargmann-Fock spaces, and a scale of weighted Besov-Sobolev spaces of holomorphic functions that includes weighted Bergman spaces of general domains as well as the Hardy space and the Dirichlet space. We apply the compactness criteria to characterize the compact Toeplitz operators on the Bergman space, deduce the compactness of Hankel operators on the Hardy space, and obtain general umbrella theorems.