On pointwise products of symmetric quasi Banach spaces and applications

TL;DR

This paper investigates properties of pointwise products and interpolation spaces of symmetric quasi Banach spaces in noncommutative settings, establishing equalities among various constructions and extending results to noncommutative Hardy spaces.

Contribution

It proves equalities among different noncommutative symmetric quasi Banach space constructions and extends classical results to noncommutative Hardy spaces.

Findings

Equalities among interpolation and product spaces in noncommutative symmetric quasi Banach spaces.

Extension of classical theorems to noncommutative symmetric quasi Hardy spaces.

Establishment of the real case of Peter Jones' theorem in noncommutative setting.

Abstract

Let $E_1,\;E_2$ be symmetric quasi Banach function spaces on $(0,\alpha)\;(0<\alpha\le\8)$. We study some properties of several constructions (the products $E_1(\M)\odot E_2(\M)$, the Calder$\rm\acute{o}$n spaces $E_1(\M)^\theta E_2(\M)^{1-\theta}$, the complex interpolation spaces $(E_1(\M),E_2(\M))_\theta$, the real interpolation method $(E_1(\M),E_2(\M))_{\theta,p}$) in the context of noncommutative symmetric quasi Banach spaces. Under some natural assumptions, we prove $$ (E_1(\M), E_2(\M))_\theta=E_1(\M)^\theta E_2(\M)^{1-\theta}=E_1^{(\frac{1}{\theta})}(\M)\odot E_2^{(\frac{1}{1-\theta})}(\M)\;(0<\theta<1). $$ As application, we extend these result to the noncommutative symmetric quasi Hardy spaces case. We also obtained the real case of Peter Jones' theorem for noncommutative symmetric quasi Hardy spaces.