A Survey on Invariant Spaces of Holomorphic Functions on Symmetric Domains

TL;DR

This survey reviews classical and recent results on invariant spaces of holomorphic functions on symmetric domains, covering various function spaces and their properties in different realizations.

Contribution

It provides a comprehensive overview of invariant holomorphic function spaces on symmetric domains, including new insights on their structure and classifications.

Findings

Includes classical and new results on invariant spaces.

Discusses minimal and maximal invariant spaces in Banach and semi-Banach classes.

Covers a range of function spaces like Bergman, Hardy, Dirichlet, Besov, and Bloch.

Abstract

We present some old and new results on a class of invariant spaces of holomorphic functions on symmetric domains, both in their circular bounded realizations and in their unbounded realizations as Siegel domains of type II. These spaces include: weighted Bergman spaces; the Hardy space $H^2$; the Dirichlet space; holomorphic Besov spaces; the Bloch space. Our main focus will be on invariant Hilbert and semi-Hilbert spaces, but we shall also discuss minimal and maximal spaces in suitable classes of invariant Banach and semi-Banach spaces.