The main purpose of this paper is to investigate the relationship between the two order structures of constructing operators for a generalized Riesz system and ( D , ℰ )-quasi bases for two fixed biorthogonal sequences {ϕn} and {Ψn}. In a previous paper, we have studied the order structure of the set Cϕ of all constructing operators for a generalized Riesz system {ϕn}, and furthermore we have shown that the notion of generalized Riesz systems has a close relation with that of ( D , ℰ )-quasi bases. For this reason, in this paper we define an order structure in the set M ϕ , ψ of all pairs of dense subspaces D and E in ℋ such that {ϕn} and {Ψn} are ( D , ℰ )-quasi bases, and shall investigate the relationships between the order sets Cϕ, CΨ and Mϕ,Ψ. These results seem to be useful to find suitable constructing operators for each physical model.

中文翻译：

---

(D, ℰ)-准基的阶结构和广义 Riesz 系统的构造算子

本文的主要目的是研究广义 Riesz 系统的构造算子的二阶结构与两个固定双正交序列 {ϕn} 和 {Ψn} 的 ( D , ℰ )-拟基之间的关系。在之前的一篇论文中，我们研究了广义 Riesz 系统 {ϕn} 的所有构造算子的集合 Cϕ 的阶结构，此外我们已经证明了广义 Riesz 系统的概念与 ( D , ℰ )-准基。为此，在本文中，我们在 ℋ 中的所有稠密子空间 D 和 E 对的集合 M ϕ , ψ 中定义了一个阶结构，使得 {ϕn} 和 {Ψn} 是 ( D , ℰ )-拟基，并且应调查阶数集 Cϕ, CΨ 和 Mϕ,Ψ 之间的关系。这些结果似乎有助于为每个物理模型找到合适的构造算子。