Noncommutative Khintchine and Paley inequalities via generic factorization

TL;DR

This paper introduces a new proof technique for inequalities involving Rademacher series and lacunary coefficients in Schatten class spaces, extending classical scalar results to operator-valued functions.

Contribution

It provides a novel proof method for noncommutative inequalities, extending scalar inequalities to Schatten class-valued functions and series.

Findings

Reproves an inequality for Rademacher series with Schatten class coefficients.

Extends Paley's theorem to Schatten class-valued lacunary series.

Introduces a unified proof approach for noncommutative inequalities.

Abstract

We reprove an inequality for Rademacher series with coefficients in the Schatten class $S_1$. Our method yields the same estimate for coefficients after suitable gaps in $S_1$-valued trigonometric series; this was known for scalar-valued functions. A very similar method gives a new proof of the extension to $S_1$-valued $H^1$ functions of Paley's theorem about lacunary coefficients.