Primarity of direct sums of Orlicz spaces and Marcinkiewicz spaces

TL;DR

This paper establishes conditions under which infinite direct sums of Orlicz and Marcinkiewicz sequence spaces are primary Banach spaces, expanding the class of known primary spaces using recent factorization theory advances.

Contribution

It provides new criteria ensuring the primarity of direct sums of Orlicz and Marcinkiewicz spaces, leveraging recent developments in factorization of the identity.

Findings

Infinite direct sums of these spaces are primary under certain conditions.

The results extend the class of known primary Banach spaces.

Uses recent factorization theory to derive these conditions.

Abstract

Let $\mathbb{Y}$ be either an Orlicz sequence space or a Marcinkiewicz sequence space. We take advantage of the recent advances in the theory of factorization of the identity carried on in [R. Lechner, Subsymmetric weak* Schauder bases and factorization of the identity, arXiv:1804.01372 [math.FA]] to provide conditions on $\mathbb{Y}$ that ensure that, for any $1\le p\le\infty$, the infinite direct sum of $\mathbb{Y}$ in the sense of $\ell_p$ is a primary Banach space, enlarging this way the list of Banach spaces that are known to be primary.