Mathematics – Functional Analysis
Scientific paper
2010-09-15
Mathematics
Functional Analysis
22 pages
Scientific paper
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 $\big(\bigoplus_{n=1}^\infty\ell_1^n \big)_{\mathrm{c}_0}$.
Laustsen Niels Jakob
Odell Edward
Schlumprecht Thomas
Zsak Andras
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