The Grothendieck Theorem in Bergman Spaces

TL;DR

The paper studies finite-dimensionality of closed subspaces $E$ of Bergman spaces $A^p(Ω)$ that embed into $A^q(Ω)$ for $1≤p<q<∞$. By viewing the natural inclusion as a composition of a bounded map and a compact embedding between Bergman spaces, the authors establish the compactness of the overall inclusion. Central tools include the Bergman inequality, Montel's theorem, and Vitali convergence with uniform integrability, enabling a short functional-analytic argument to deduce finite dimensionality. This work extends Grothendieck-type results to the setting of Bergman spaces and provides a clear criterion for when closed subspaces must be finite-dimensional."

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.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1: Bergman inequality,
• Proposition 2.2
• proof
• Theorem 2.3
• Definition 2.4: Uniform Integrability
• Theorem 2.5: Vitali Convergence Theorem,
• proof : Proof of Theorem \ref{['th:compact']}