Ordered Structures of Constructing Operators for Generalized Riesz Systems

A sequence {φ푛} in a Hilbert spaceH with inner product < ⋅, ⋅ > is called a generalized Riesz system if there exist an ONB e = {e푛} inH and a densely defined closed operator T inH with densely defined inverse such that {e푛} ⊂ D(T) ∩ D((T−1)∗) and Te푛 = φ푛 , n = 0, 1, ⋅ ⋅ ⋅ , and (e, T) is called a constructing pair for {φ푛} and T is called a constructing operator for {φ푛}. The main purpose of this paper is to investigate under what conditions the ordered set C휑 of all constructing operators for a generalized Riesz system {φ푛} hasmaximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications.


Introduction
Generalized Riesz systems can be used to construct some physical operators (non-self-adjoint Hamiltonian, generalized lowering operator, generalized raising operator, number operator, etc.) [1][2][3]. Then these operators provide a link to quasi-Hermitian quantum mechanics, and its relatives. Many researchers have investigated such operators both from the mathematical point of view and for their physical applications [4][5][6][7][8][9]. Let { 푛 } be a generalized Riesz system with a constructing pair ( , ).
휑 , 휑 , and 휑 are called the non-self-adjoint Hamiltonian, the generalized lowering operator, and the generalized raising operator for { 푛 }, respectively. The physical operators of the extended quantum harmonic oscillator and the Swanson model are of this form (see  in Section 3).
From this fact, it seems to be important to consider under what conditions biorthogonal sequences are generalized Riesz systems and in [1][2][3] we have investigated this problem. In this paper, we shall focus on the following facts: physical operators defined by a generalized Riesz system { 푛 } depend on constructing pairs; for example, their operators may not be densely defined for some constructing pairs. On the other hand, if there exists a dense subspace D in H for a constructing pair ( , ) which is a core for such that D ⊂ D, D ⊂ D, and D ⊂ D, then they have an algebraic structure; in detail, the -algebra on D is defined by the restrictions of the operators 휑 and 휑 to D [10]. Thus it seems to be important to find a constructing pair fitting to each of the physical applications. From this reason, in this paper we shall investigate the properties of constructing pairs for a generalized Riesz system.
In Section 2, we shall investigate the basic properties of constructing operators. Let { 푛 } be a generalized Riesz system with a constructing pair ( , ). The constructing operators for International Journal of Mathematics and Mathematical Sciences basis; that is, and −1 are bounded, but they are not unique in general. So, we investigate the set ,휑 of all constructing operators for . In Proposition 1, we shall show that it is possible to fix an ONB = { 푛 } in H without loss of generality for our study in this paper. Hence, we fix an ONB in H and denote ,휑 by 휑 for simplicity. We consider the following problem: Is any sequence { 푛 } which is biorthogonal to { 푛 } a generalized Riesz system?
Here we put Then we shall show in Proposition 5 that if 푁 휑 ̸ = 0, then { 푛 } is a generalized Riesz system and ( , ( −1 ) * ) is a constructing pair for { 푛 } for every ∈ 푁 휑 , and the mapping ∈ 휓 . These results seem to be useful to find fitting constructing operators for each physical model because every closed operator in H satisfying 푆 ⊂ ⊂ 퐿 belongs to 휑 , where 푆 is the smallest element of 휑 and 퐿 is the largest element of 휑 .

The Basic Properties of Constructing Operators
In this section, we shall investigate the basic properties of constructing operators. Let { 푛 } be a generalized Riesz system with a constructing pair ( , ). It is easily shown that if { 푛 } is a Riesz basis, then the constructing operator for { 푛 } is unique for (see Proposition 1 in detail). But, in general, the constructing operators for { 푛 } are not unique, and so we put By Proposition 1, we have the following.
Corollary . Let ∈ ,휑 and * = | * | be the polar decomposition of * . en fl * is an ONB in H and | * | = ∈ ,휑 . Furthermore, we have Thus we may fix an ONB = { 푛 } in H without loss of generality for investigating the properties of ,휑 , and so throughout this paper, we fix an ONB in H and denote ,휑 by 휑 for simplicity. Next we consider the following problem: To consider when this equality holds, we define the operators 0 휑, , 휑, , and ,휑 for any sequence { 푛 } in H as follows: These operators have played an important role for our studies [3] and also in this paper. By Lemma 2.1, 2.2 in [3] we have the following.
Lemma . Suppose that { 푛 } is a generalized Riesz system and ∈ 휑 . en the following statements are equivalent.
If this holds true, then is called natural.
Take an arbitrary ∈ ∪ {0}. Then, since Thus, we have This completes the proof.
We denote the set of all natural constructing operators for Then we have the following.
Proposition . Suppose that { 푛 } is a generalized Riesz system. en the following statements hold.

also a generalized Riesz system and put
en the mapping Proof. The statements (1) and (2) are easily shown.
Similarly, we can show ∈ 푁 휑 in case that 0 ⊂ and can show ∈ 푁 휓 in case that ⊂ 0 or 0 ⊂ . This completes the proof.
As for the uniqueness of constructing operators for a generalized Riesz system we have the following.
Proposition . Let { 푛 } be a generalized Riesz system. en the following statements hold.

Ordered Structures of 휑
In this section, we shall consider the ordered set 휑 of all constructing operators for a generalized Riesz system { 푛 } with order ⊂ and investigate when 휑 has maximal elements, minimal elements, the largest element, and the smallest element. The following result gives a motivation to study the ordered structures of 휑 Lemma . Suppose that , ∈ 휑 and ⊂ . en, for any linear operator such that ⊂ ⊂ , the closure of belongs to 휑 .
Proof. This is trivial.
For biorthogonal sequences satisfying density-conditions, we have the following.
Proposition . e following statements hold.
Example (the extended quantum harmonic oscillator). The Hamiltonian of this model is the non-self-adjoint operator, introduced in [11,12], We put Then, Similarly, we define a sequence 훽 fl { (훽) 푛 } in S(R) as follows: The following example is a modification of the nonself-adjoint Hamiltonian 훽 in Example 9 exchanging the momentum operator with the position operator .
All physical models discussed above are regular cases, but it seems to be mathematically meaningful to study nonregular cases and furthermore the studies may become useful for physical applications in future. Let { 푛 } be a generalized Riesz system. First we investigate under what conditions 휑 has maximal elements and minimal elements.
For a subset F of 휑 , we put Then we have the following.
Lemma . Let F be a subset of 휑 . en, 0 is a maximal (resp., minimal, the largest and the smallest) element of F if and only if ( −1 0 ) * is a minimal (resp., maximal, the smallest and the largest) element of F 휓 .
Proof. Suppose that 0 is a maximal element of F. Take an arbitrary ∈ F 휓 satisfying ⊂ ( −1 0 ) * . Then we have that = ( −1 ) * for some ∈ F and 0 ⊂ , which implies by the maximality of 0 that = 0 and = ( −1 0 ) * . Thus, ( −1 0 ) * is a minimal element of F 휓 . Furthermore, we can similarly show that if ( −1 0 ) * is a minimal element of F 휓 , then 0 is a maximal element F. The other statements are similarly shown.
eorem . Let F be a subset of 휑 . en we have the following: (1) e following statements are equivalent: (i) F has a maximal element. (ii) ere exists a closed operator in H such that ⊂ for all ∈ F. (iii) F 휓 has a minimal element.
(2) e following statements are equivalent: (iii) F 휓 has a maximal element and a minimal element.
(ii) ⇒(i) Suppose that there exists a closed operator in H such that ⊂ for all ∈ F. Then for any totally ordered subset of F we have ( * ) ⊂ ⋂ 푇∈퐶 ( * ). Hence, it follows that ⋂ 푇∈퐶 ( * ) is dense in H, which implies by Lemma 12 that F has an upper bounded element. By Zorn's lemma, F has a maximal element.
(ii) ⇒(i) Suppose that there exists a closed operator in H such that ( −1 ) * ⊂ for all ∈ F. Then we can similarly show that F 휓 has a maximal element, which implies by Lemma 13 that F has a minimal element.
We remark that the closed operators and in Theorem 14 do not need any other conditions, for example, the existence of inverse.
By Theorem 14, we have the following.
Corollary . Let 0 ∈ 휑 and put F 푇 0 = { ∈ 휑 ; 0 ⊂ }. en the following statements hold. ( ) Suppose that there exists a closed operator in H such that ⊂ for all ∈ F 푇 0 . en there exists a maximal element of 휑 which is an extension of 0 .
( ) Suppose that there exists a closed operator in H such that ( −1 ) * ⊂ for all ∈ F 푇 0 . en there exists a minimal element of 휑 which is a restriction of 0 .
Proof. (1) By Theorem 14, F 푇 0 has a maximal element 1 . Here we show that 1 is a maximal element of 휑 . Indeed, this follows since ∈ F 푇 0 for any element of 휑 satisfying 1 ⊂ . We can similarly show (2).
Next we investigate the existence of the smallest element and of the largest element of 휑 .
For a generalized Riesz system { 푛 }, suppose that there exist the largest element 퐿 of 휑 and the smallest element 푆 of 휑 . Then every closed operator in H satisfying 푆 ⊂ ⊂ 퐿 is a constructing operator of { 푛 }, and so we can construct all kinds of non-self-adjoint Hamiltonians −1 , lowering operator −1 and raising operator −1 for { 푛 }. It may be possible to find constructing operators suitable for each of the physical models.

Conclusions
All the results presented in this paper are of pure mathematical nature, but we hope that they will be applied to more physical models in future. For example, we argue that cases like the CCR-algebras and their physical applications could probably studied by taking suitable constructing operators for convenient generalized Riesz systems.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.