On pointwise products of symmetric quasi Banach spaces and applications

Let $E_1,E_2$ be symmetric quasi Banach function spaces on $(0,\alpha )(0<\alpha \le \text{8})$. We study some properties of several constructions (the products $E_1(\text{M})\odot E_2(\text{M})$, the Calder$\acute{\mathrm{o}}$n spaces $E_1(\text{M}{)}^{\theta }{E}_2(\text{M}{)}^{1-\theta }$, the complex interpolation spaces $(E_1(\text{M}),E_2(\text{M}){)}_{\theta }$, the real interpolation method $(E_1(\text{M}),E_2(\text{M}){)}_{\theta ,p}$) in the context of noncommutative symmetric quasi Banach spaces. Under some natural assumptions, we prove $$(E_1(\text{M}),E_2(\text{M}){)}_{\theta }=E_1(\text{M}{)}^{\theta }{E}_2(\text{M}{)}^{1-\theta }=E_1^{(\frac{1}{\theta })}(\text{M})\odot E_2^{(\frac{1}{1-\theta })}(\text{M})(0<\theta <1)\mathrm{.}$$ 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.