On products of noncommutative symmetric quasi Banach spaces and applications

Let E1,E2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1,\;E_2$$\end{document} be symmetric quasi Banach function spaces on (0,α)(0<α≤∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\alpha )\;(0<\alpha \le \infty )$$\end{document}. We study some properties of several constructions (the products E1(M)⊙E2(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1({\mathcal {M}})\odot E_2({\mathcal {M}})$$\end{document}, the Calderón spaces E1(M)θE2(M)1-θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1({\mathcal {M}})^\theta E_2({\mathcal {M}})^{1-\theta }$$\end{document}, the complex interpolation spaces (E1(M),E2(M))θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(E_1({\mathcal {M}}),E_2({\mathcal {M}}))_\theta $$\end{document}, the real interpolation method (E1(M),E2(M))θ,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(E_1({\mathcal {M}}),E_2({\mathcal {M}}))_{\theta ,p}$$\end{document}) in the context of noncommutative symmetric quasi Banach spaces. Under some natural assumptions, we prove (E1(M),E2(M))θ=E1(M)θE2(M)1-θ=E11θ(M)⊙E211-θ(M)(0<θ<1).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (E_1({\mathcal {M}}), E_2({\mathcal {M}}))_\theta =E_1({\mathcal {M}})^\theta E_2({\mathcal {M}})^{1-\theta }=E_1^{\left( \frac{1}{\theta }\right) }({\mathcal {M}})\odot E_2^{\left( \frac{1}{1-\theta }\right) }({\mathcal {M}})\;(0<\theta <1). \end{aligned}$$\end{document}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.