Unconditional basic sequences in function spaces with applications to Orlicz spaces

TL;DR

This paper establishes conditions under which certain function spaces behave like $L_p$-spaces regarding their unconditional bases, leading to a classification of symmetric basic sequences with applications to Orlicz spaces.

Contribution

It introduces criteria for function spaces to mimic $L_p$-space behavior, enabling classification of their basic sequences, especially in Orlicz spaces.

Findings

Unconditional bases in these spaces are either equivalent to $ ext{ell}_2$ bases or disjointly supported sequences.

The classification of symmetric basic sequences in the studied spaces is achieved.

Applications to Orlicz spaces demonstrate the practical relevance of the theoretical results.

Abstract

We find conditions on a function space $\bf{L}$ that ensure that it behaves as an $L_p$-space in the sense that any unconditional basis of a complemented subspace of $\bf{L}$ either is equivalent to the unit vector system of $\ell_2$ or has a subbasis equivalent to a disjointly supported basic sequence. This dichotomy allows us to classify the symmetric basic sequences of $\bf{L}$. Several applications to Orlicz function spaces are provided.