The factorization property of $\ell^\infty(X_k)$

TL;DR

This paper investigates the conditions under which the identity operator on an $oldsymbol{ ext{ell}}^ extbf{ ext{infty}}$-sum of Banach spaces factors through a bounded linear operator with a large diagonal.

Contribution

It establishes general conditions ensuring the identity operator factors through a bounded operator with a large diagonal on the $oldsymbol{ ext{ell}}^ extbf{ ext{infty}}$-sum of Banach spaces.

Findings

Identifies conditions for the identity to factor through operators with large diagonals.

Provides a framework for understanding factorization in $oldsymbol{ ext{ell}}^ extbf{ ext{infty}}$-sums.

Extends previous results on operator factorization in Banach space theory.

Abstract

In this paper we consider the following problem: Let $X_k$, be a Banach space with a normalized basis $(e_{(k,j)})_j$, whose biorthogonals are denoted by $(e_{(k,j)}^*)_j$, for $k\in\mathbb{N}$, let $Z=\ell^\infty(X_k:k\in\mathbb{N})$ be their $\ell^\infty$-sum, and let $T:Z\to Z$ be a bounded linear operator, with a large diagonal, i.e. $$\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.$$ Under which condition does the identity on $Z$ factor through $T$? The purpose of this paper is to formulate general conditions for which the answer is positive.