The Grothendieck Theorem in Bergman Spaces

TL;DR

This paper proves that certain subspaces of holomorphic L^p spaces that are also contained in higher L^q spaces must be finite-dimensional, extending the Grothendieck theorem to Bergman spaces.

Contribution

It establishes a new finite-dimensionality result for subspaces in Bergman spaces under specific integrability conditions, extending classical theorems.

Findings

Subspaces contained in both L^p and L^q spaces are finite-dimensional.

The result applies to closed subspaces of holomorphic L^p spaces.

Extension of the Grothendieck theorem to Bergman spaces.

Abstract

In this paper, we prove that if $E$ is a closed subspace of the holomorphic $L^p$-integrable space and is also contained in the holomorphic $L^q$-integrable space, for any $p > 1$ and any $q > p$, then the dimension of $E$ must be finite.