#### Abstract

A sequence in a Hilbert space with inner product is called a generalized Riesz system if there exist an ONB in and a densely defined closed operator in with densely defined inverse such that and , , and is called a constructing pair for and is called a constructing operator for . The main purpose of this paper is to investigate under what conditions the ordered set of all constructing operators for a generalized Riesz system has maximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications.

#### 1. 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.) [1â€“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 [4â€“9]. Let be a generalized Riesz system with a constructing pair . Then, putting , and are biorthogonal sequences, that is, , . For any we can define the operators: , , and , where , , and are standard self-adjoint Hamiltonian, lowering operator, and raising operator for , respectively, where for , , . Since, , and are called the non-self-adjoint Hamiltonian, the generalized lowering operator, and the generalized raising operator for , respectively. The physical operators of the extended quantum harmonic oscillator and the Swanson model are of this form (see Examples 9â€“11 in Section 3).

From this fact, it seems to be important to consider under what conditions biorthogonal sequences are generalized Riesz systems and in [1â€“3] we have investigated this problem. In this paper, we shall focus on the following facts: physical operators defined by a generalized Riesz system 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 in for a constructing pair which is a core for such that , , and , then they have an algebraic structure; in detail, the -algebra on is defined by the restrictions of the operators and to [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 be a generalized Riesz system with a constructing pair . The constructing operators for are unique for the fixed ONB in if is a Riesz basis; that is, and are bounded, but they are not unique in general. So, we investigate the set of all constructing operators for . In Proposition 1, we shall show that it is possible to fix an ONB in without loss of generality for our study in this paper. Hence, we fix an ONB in and denote by for simplicity. We consider the following problem:* Is any sequence ** which is biorthogonal to ** a generalized Riesz system?*

Here we putThen we shall show in Proposition 5 that if , then is a generalized Riesz system and is a constructing pair for for every , and the mappingis a bijection, where is the set of all constructing operators for andFurthermore, we shall show in Proposition 6 that if there exists a bounded operator in , then and .

In Section 3, we shall consider the ordered set with order and investigate under what conditions the ordered set has a maximal element, a minimal element, the smallest element, and the largest element. First we have shown that if linear span is dense in , then and there exists the smallest element of , and furthermore if and is dense in , there exist the smallest element of and the largest element of , and in particular, if and are regular biorthogonal sequences in , that is, both and are dense in , then , , and has the smallest element and the largest element. Next we shall consider the case when is not necessarily dense in . In Theorem 14, we shall show that for a subset of if there exists a closed operator in such that for all , then has a maximal element, and furthermore, if there exists a closed operator in such that for all , then have a maximal element and a minimal element.

For the existence of the smallest element of and of the largest element of , we shall show in Theorem 16 that if there exist closed operators and in such that and for all , then has the smallest element and the largest element. Furthermore, for a biorthogonal pair of generalized Riesz systems satisfying and , we shall show in Theorem 18 that and have the smallest element and the largest element, respectively, if and only if there exist closed operators and in such that and for all and . These results seem to be useful to find fitting constructing operators for each physical model because every closed operator in satisfying belongs to , where is the smallest element of and is the largest element of .

#### 2. The Basic Properties of Constructing Operators

In this section, we shall investigate the basic properties of constructing operators. Let be a generalized Riesz system with a constructing pair . It is easily shown that if is a Riesz basis, then the constructing operator for is unique for (see Proposition 1 in detail). But, in general, the constructing operators for are not unique, and so we putFirst, we investigate the relationship between and for the other ONB in .

Proposition 1. *Let and be any ONB in . Then the following statements hold.**(1) is a constructing pair for , where is a unitary operator on defined by , and**(2) For the non-self-adjoint Hamiltonian, the generalized lowering operator, and the generalized raising operator for , we have*

*Proof. *(1) This is almost trivial.

(2) This follows from

By Proposition 1, we have the following.

Corollary 2. *Let and be the polar decomposition of . Then is an ONB in and . Furthermore, we have*

Thus we may fix an ONB in without loss of generality for investigating the properties of , and so throughout this paper, we fix an ONB in and denote by for simplicity. Next we consider the following problem:* Suppose that ** is a biorthogonal pair such that ** is a generalized Riesz system. Then, is ** also a generalized Riesz system?*

Let and . Then is a biorthogonal pair and is a generalized Riesz system with a constructing pair . If , then is a generalized Riesz system with a constructing pair . But, the equality does not necessarily hold. To consider when this equality holds, we define the operators , , and for any sequence in as follows:

These operators have played an important role for our studies [3] and also in this paper. By emma 2.1, 2.2 in [3] we have the following.

Lemma 3. *(1) and are densely defined linear operators in such that**(2) and .**(3) is closable if and only if is closable if and only if is dense in . If this holds, then*

From now on, let be a biorthogonal pair.

Lemma 4. *Suppose that is a generalized Riesz system and . Then the following statements are equivalent. *(i)*.*(ii)*.** If this holds true, then is called natural.*

*Proof. *(i)(ii) This is trivial.

(ii)(i) By definition of , we have . Furthermore, by Lemma 3, (2), we haveTake an arbitrary . Then, sincefor , we have and . Hence it follows from (13) thatThus, we haveThis completes the proof.

We denote the set of all natural constructing operators for by ; that is,Then we have the following.

Proposition 5. *Suppose that is a generalized Riesz system. Then the following statements hold.**(1) If , then is a generalized Riesz system and is a constructing pair for for every .**(2) Suppose that is also a generalized Riesz system and putThen the mappingis a bijection.**(3) Suppose that and satisfying or . Then . Similarly, suppose that and satisfying or . Then .*

*Proof. *The statements (1) and (2) are easily shown.

(3) Suppose that . Then, since , it follows that , which implies that . Similarly, we can show in case that and can show in case that or . This completes the proof.

As for the uniqueness of constructing operators for a generalized Riesz system we have the following.

Proposition 6. *Let be a generalized Riesz system. Then the following statements hold.**(1) Suppose that is a Riesz basis, then is also a Riesz basis, and .**(2) Suppose that and are dense in . Then we have the following.**(i) If there exists an element of such that is bounded, then and .**(ii) If there exists an element of such that is bounded, then and .*

*Proof. *(1) Since is a Riesz basis, there exists an element of such that and are bounded, which implies that and for all .

(2) (i) Since and is bounded, we have . Take an arbitrary . Then, since and is bounded, we have . Thus, . We show . Take an arbitrary . Since and is bounded, it follows that , which implies . Thus, .

(ii) This is similarly shown.

#### 3. Ordered Structures of

In this section, we shall consider the ordered set of all constructing operators for a generalized Riesz system with order and investigate when has maximal elements, minimal elements, the largest element, and the smallest element. The following result gives a motivation to study the ordered structures of

Lemma 7. *Suppose that and . Then, for any linear operator such that , the closure of belongs to .*

*Proof. *This is trivial.

For biorthogonal sequences satisfying density-conditions, we have the following.

Proposition 8. *The following statements hold.**(1) Suppose that is dense in . Then, is a generalized Riesz system and , and is the smallest element of . Furthermore, suppose that is dense in . Then, is the largest element of .**(2) Suppose that is dense in . Then, is a generalized Riesz system and , and is the smallest element in . Furthermore, suppose that is dense in . Then, is the largest element in .**(3) Suppose that is regular; that is, both and are dense in . Then, and are generalized Riesz systems and and , and is the smallest element in , is the smallest element in , is the largest element in , and is the largest element in .*

*Proof. *(1) We can show using Lemma 3 that is a generalized Riesz system with a constructing pair and the constructing operator for is the smallest element in . For more detail, refer to [3]. Furthermore, a sequence which is biorthogonal to is unique. In fact, let and be any sequences in which are biorthogonal to . Then, since for and is dense in , we have for every . We show . Take an arbitrary . Then, is biorthogonal to . By the uniqueness of biorthogonal sequences to , we have , which implies that and . Suppose that and are dense in . We show that is the largest element in . Since , is a densely defined closed operator in , and since , it has a densely defined inverse . Furthermore, since , . Thus we have . Since , it follows that . Next we show that is the largest element in . Take an arbitrary . Then , and so , . Hence we have . Thus , and so is the largest element in .

(2) This is proved at the same way as (1).

(3) Since and , it follows that and are dense in , which implies by (1) and (2) that the statement (3) holds.

Here we give some physical examples. Let , be an ONB in consisting of the Hermite functions which is contained in the Schwartz space of all infinitely differential rapidly decreasing functions on . We define the moment operator and the position operator byandThen and are self-adjoint operators in and is a core for and , and furthermore and , and on . Next we define the standard bosonic operators , byThen,and on .

*Example 9 (the extended quantum harmonic oscillator). *The Hamiltonian of this model is the non-self-adjoint operator, introduced in [11, 12],We putThen, and , where is a unitary operator defined by . Hence we can define a sequence in bySimilarly, we define a sequence in as follows:Then and are regular biorthogonal sequences in which are generalized Riesz systems with constructing pairs and , respectively, andBy Proposition 8, is the smallest constructing operator and is the largest constructing operator for and . Similarly, is the smallest constructing operator and is the largest constructing operator for and .

The following example is a modification of the non-self-adjoint Hamiltonian in Example 9 exchanging the momentum operator with the position operator .

*Example 10. *We introduce a non-self-adjoint HamiltonianWe define sequences and in as follows:andThen and are regular biorthogonal sequences in which are generalized Riesz systems with constructing pairs and , respectively, where and , and , , , and .

*Example 11 (the Swanson model). *The Swanson Hamiltonian, introduced in [11, 13], is a non-self-adjoint HamiltonianWe define sequences and in as follows:andwhere , , and and are constants satisfying . Then and are regular biorthogonal sequences in contained in which are generalized Riesz systems with constructing operators and , respectively. For the generalized lowering operator and the raising operator , we haveBy Proposition 8, (resp., ) is the smallest constructing operator and(resp., ) is the largest constructing operator for (resp., ) and every closed operator(resp., ) in satisfying (resp., ) is a constructing operator for (resp., ).

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 be a generalized Riesz system. First we investigate under what conditions has maximal elements and minimal elements.

Let be a totally ordered subset of . Then, it is easily shown that , , for any . Hence we may putWe have the following statements.

Lemma 12. *Let be any totally ordered subset of . The following statements hold.**(1) Suppose that is dense in . Then there exists an upper bounded element of .**(2) Suppose that is dense in . Then there exists a lower bounded element of .**(3) Suppose that and are dense in . Then for every linear operator such that , the closure of belongs to .*

*Proof. *(1) We putwhere is an operator in whose domain contains . Since is totally ordered, it follows that is a subspace in and for any operators , in whose domains contain . Hence, does not depend on the method of choosing whose domain contains . Thus is a well-defined densely defined linear operator in such that for all . We show that is closable. Indeed, we may showwhere whose domain contains . Take an arbitrary . Then, we havefor all , where whose domain contains . Hence, and . Since for all , is trivial. Thus, (38) holds. By (38) and the assumption of (1), is dense in , that is, is closable. Next we show that has a densely defined inverse. Suppose that , . Then for some , and so since has an inverse. Thus has an inverse. Since and is dense in for all , it follows that the inverse of is densely defined, which implies that the closure of has a densely defined closed operator in such that for all . Finally we showClearly, for all . Next we show that for any there exists an element of such thatfor all . Indeed, take an arbitrary . Since , there exists an element of such that . Let any . Since is totally ordered, either or holds. Suppose that . Since , it follows that , andwhich implies that since has inverse. The equality is similarly shown in case that . Hence, we have that and for all . Thus (41) holds. By (38) and (41) we have . Furthermore, we have , for all . Thus we have and is an upper bounded element of .

(2) We putwhere is any element of . Since and for all , is a well-defined densely defined closed operator in such that for all . Hence, it is sufficient to show . Since for all and has the inverse, has the inverse. Furthermore, we may showIn fact, take an arbitrary . Since is totally ordered, there exists an element of such that for all . Hence, . The inverse inclusion is clear. Hence (44) holds. By the assumption and (44), is dense in . Furthermore, since for all , we have and . Since , it follows that . Thus, and it is a lower bound of .

(3) This follows from (1) and (2).

For a subset of , we putThen we have the following.

Lemma 13. *Let be a subset of . Then, is a maximal (resp., minimal, the largest and the smallest) element of if and only if is a minimal (resp., maximal, the smallest and the largest) element of .*

*Proof. *Suppose that is a maximal element of . Take an arbitrary satisfying