Dichotomy theorems for random matrices and closed ideals of operators on \((\bigoplus _{n=1}^{\infty }\ell _{1}^{n})_{c_o}\)
The authors prove two dichotomy theorems about sequences of operators into \(L_{1}\) given by random matrices. These results provide information on the closed ideal structure of the Banach algebra of all operators on the space \(\left( \bigoplus_{n=1}^{\infty}\ell_{1}^{n}\right) _{c_{0}}.\) One of the main results of the paper is the following theorem:NEWLINENEWLINEFor each positive integer \(m\), let \(T^{(m)}:\ell_{\infty}^{m}\left( \ell _{1}^{m}\right) \rightarrow L_{1}\) be an operator such that the entries of the corresponding random matrix \(\left( T_{i,j}^{(m)}\right) \) form a sequence of independent, symmetric random variables with NEWLINE\[NEWLINE \left\| T^{(m)}\right\| =\max\left\{ \mathbb{E}\left|\sum_{i=1}^{m} T_{i,j_{i}}^{(m)}\right| :j_{1},\dots,j_{m}\in\{1,\dots,m\} \right\} \leq1. NEWLINE\]NEWLINE ThenNEWLINENEWLINE(i) either the identity operators \(Id_{\ell_{1}^{k}}\) (\(k\in\mathbb{N}\)) uniformly factor through the \(T^{(m)},\)NEWLINENEWLINE(ii) or the \(T^{(m)}\) uniformly approximately factor through \(\ell_{\infty }^{k}\) (\(k\in\mathbb{N}\)).
scientific article; zbMATH DE number 6073817