Almost-compact embeddings

Abstract

We study almost-compact embeddings between Banach function spaces. We prove a necessary and sufficient condition in terms of almost-everywhere convergence.

We also study the dependence of an almost-compact embedding on the measure space. We introduce a certain product operator and show its intimate relation to an almost-compact embedding.

We also characterize general almost-compact embeddings among Lorentz and Marcinkiewicz endpoint spaces.

Keywords

• Almost-compact embedding
• Banach function space
• rearrangement-invariant space
• product operator
• almost-everywhere convergence
• Lorentz endpoint space
• Marcinkiewicz endpoint space