On pointwise products of symmetric quasi Banach spaces and applications

Abstract

Let $E_1,E_2$ be symmetric quasi Banach function spaces on $(0,α)\;(0<α\le\8)$. We study some properties of several constructions (the products $E_1(\M)\odot E_2(\M)$, the Calder$\acute{\mathrm{o}}$n spaces $E_1(\M)^θE_2(\M)^{1-θ}$, the complex interpolation spaces $(E_1(\M),E_2(\M))_θ$, the real interpolation method $(E_1(\M),E_2(\M))_{θ,p}$) in the context of noncommutative symmetric quasi Banach spaces. Under some natural assumptions, we prove $$ (E_1(\M), E_2(\M))_θ=E_1(\M)^θE_2(\M)^{1-θ}=E_1^{(\frac{1}θ)}(\M)\odot E_2^{(\frac{1}{1-θ})}(\M)\;(0<θ<1). $$ 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.

19 pades

References in corpus (2)

• Riesz and Szegö type factorizations for noncommutative Hardy spaces
• Complex interpolation of noncommutative Hardy spaces associated semifinite von Neumann algebra