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

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 $(e_{(k,j)}^{\ast }{)}_j$, 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. $$\underset{k,j}{\inf }|e_{(k,j)}^{\ast }(T(e_{(k,j)})|>0.$$ 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.

27 pages, 1 figure