Dichotomy theorems for random matrices and closed ideals of operators on $\big(\bigoplus_{n=1}^\infty\ell_1^n \big)_{\mathrm{c}_0}$

TL;DR

This paper establishes two dichotomy theorems for sequences of operators into L_1 via random matrices, revealing their factorization properties and implications for the ideal structure of operators on a specific Banach space.

Contribution

It introduces new dichotomy theorems for operator sequences involving random matrices, advancing understanding of their factorization and the ideal structure of operator algebras.

Findings

Operators either uniformly factor identity on ^k or approximate factor through c_0.

Results apply to operators on arbitrary Banach space sequences.

Provides insights into the ideal structure of the operator algebra on a specific Banach space.

Abstract

We prove two dichotomy theorems about sequences of operators into $L_1$ given by random matrices. In the second theorem we assume that the entries of each random matrix form a sequence of independent, symmetric random variables. Then the corresponding sequence of operators either uniformly factor the identity operators on $\ell_1^k$ $(k\in\mathbb N$) or uniformly approximately factor through $\mathrm{c}_0$. The first theorem has a slightly weaker conclusion still related to factorization properties but makes no assumption on the random matrices. Indeed, it applies to operators defined on an arbitrary sequence of Banach spaces. These results provide information on the closed ideal structure of the Banach algebra of all operators on the space $\big(\bigoplus_{n=1}^\infty\ell_1^n \big)_{\mathrm{c}_0}$.