Let \(E_1,\;E_2\) be symmetric quasi Banach function spaces on \((0,\alpha )\;(0<\alpha \le \infty )\). We study some properties of several constructions (the products \(E_1({\mathcal {M}})\odot E_2({\mathcal {M}})\), the Calderón spaces \(E_1({\mathcal {M}})^\theta E_2({\mathcal {M}})^{1-\theta }\), the complex interpolation spaces \((E_1({\mathcal {M}}),E_2({\mathcal {M}}))_\theta \), the real interpolation method \((E_1({\mathcal {M}}),E_2({\mathcal {M}}))_{\theta ,p}\)) in the context of noncommutative symmetric quasi Banach spaces. Under some natural assumptions, we prove

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.

令\（E_1，\; E_2 \）是\（（0，\ alpha）\;（0 <\ alpha \ le \ infty）\）上的对称拟Banach函数空间。我们研究了几种构造（产品\（E_1（{\ mathcal {M}}）\ odot E_2（{\ mathcal {M}}）\），Calderón空间\（E_1（{\ mathcal {M} }）^ \ theta E_2（{\ mathcal {M}}）^ {1- \ theta} \），复数插值空间\（（E_1（{\ mathcal {M}}），E_2（{\ mathcal {M }}））_ \ theta \），其中的实插值方法\（（E_1（{\ mathcal {M}}），E_2（{\ mathcal {M}}））_ {\ theta，p} \））非交换对称拟Banach空间的上下文。在一些自然假设下，我们证明

作为应用，我们将这些结果扩展到非交换对称拟Hardy空间情况。我们还获得了非交换对称拟Hardy空间的Peter Jones定理的实例。