In this paper we consider the following problem: let Xk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by ${(e_{(k,j)}^*)_j}$ , for $k\in\N$ , let $Z=\ell^\infty(X_k:k\kin\N)$ be their l∞-sum, and let $T:Z\to Z$ be a bounded linear operator with a large diagonal, i.e., $$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$ 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.

中文翻译：

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l∞(Xk) 的因式分解性质

在本文中，我们考虑以下问题：让Xķ, 是具有归一化基的 Banach 空间 (e(k, j))j，其双正交表示为${(e_{(k,j)}^*)_j}$， 为了$k\in\N$， 让$Z=\ell^\infty(X_k:k\kin\N)$成为他们的l∞-sum，并让$T:Z\to Z$是具有大对角线的有界线性算子，IE,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{对齐*}$$身份在什么条件下Z因素通过吨? 本文的目的是制定答案为肯定的一般条件。