Noncommutative Burkholder/Rosenthal inequalities II: applications

TL;DR

This paper extends noncommutative Burkholder/Rosenthal inequalities to provide norm estimates for sums of independent random variables in noncommutative $L_p$-spaces, with applications to random matrices and subspace characterizations.

Contribution

It generalizes classical inequalities to the noncommutative setting and applies these results to analyze random matrices and subspace structures in noncommutative $L_p$-spaces.

Findings

Derived norm equivalences for singular values of random matrices with independent entries.

Characterized subspaces and ideals realizable as subspaces of noncommutative $L_p$-spaces for $2<p< finite.

Abstract

We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<\infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the $p$-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative $L_p$ for $2<p<\infty$.