# Ordered Structures of Constructing Operators for Generalized Riesz Systems

Authors: Hiroshi Inoue
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

A sequence{φn} in a Hilbert space H with inner product <·,·> is called a generalized Riesz system if there exist an ONB e={en} in H and a densely defined closed operator T in H with densely defined inverse such that {en}⊂D(T)∩D((T-1)⁎) and Ten=φn, n=0,1,⋯, and (e,T) is called a constructing pair for {φn} and T is called a constructing operator for {φn}. The main purpose of this paper is to investigate under what conditions the ordered set Cφ of all constructing operators for a generalized Riesz system {φn} 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.


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## Body

## 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.) [1–3]. Then these operators provide a link toquasi-Hermitian quantum mechanics, and its relatives. Many researchers have investigated such operators both from the mathematical point of view and for their physical applications [4–9]. Let φn be a generalized Riesz system with a constructing pair (e,T). Then, putting ψnT=(T-1)∗en,n=0,1,⋯, φn and ψnT are biorthogonal sequences, that is, <φn,ψmT>=δnm, n,m=0,1,⋯. For any αn⊂C we can define the operators: Hφα≔THeαT-1, Aφα≔TAeαT-1, and Bφα≔TBeαT-1, where Heα≔∑n=0∞αnen⊗e¯n, Aeα≔∑n=0∞αn+1en+1⊗e¯n, and Beα≔∑n=0∞αn+1en+1⊗e¯n are standard self-adjoint Hamiltonian, lowering operator, and raising operator for en, respectively, where for x,y∈H, (x⊗y)ξ≔<ξ,y>x, ξ∈H. Since(1)Hφαφn=αnφn,Aφαφn=0,n=0αnφn-1,n=1,2,⋯Bφαφn=αn+1φn+1,n=0,1,⋯,Hφα, Aφα, and Bφα are called the non-self-adjoint Hamiltonian, the generalized lowering operator, and the generalized raising operator for φn, respectively. The physical operators of the extended quantum harmonic oscillator and the Swanson model are of this form (see Examples 9–11 in Section 3).From this fact, it seems to be important to consider under what conditions biorthogonal sequences are generalized Riesz systems and in [1–3] we have investigated this problem. In this paper, we shall focus on the following facts: physical operators defined by a generalized Riesz system φn 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 (e,T) which is a core for T such that HeαD⊂D, AeαD⊂D, and BeαD⊂D, then they have an algebraic structure; in detail, the O-algebra on D is defined by the restrictions of the operators Aφα and Bφα 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 Section2, we shall investigate the basic properties of constructing operators. Let φn be a generalized Riesz system with a constructing pair (e,T). The constructing operators for φn are unique for the fixed ONB e in H if φn is a Riesz basis; that is, T and T-1 are bounded, but they are not unique in general. So, we investigate the set Ce,φ of all constructing operators for e. In Proposition 1, we shall show that it is possible to fix an ONB e=en in H without loss of generality for our study in this paper. Hence, we fix an ONB e in H and denote Ce,φ by Cφ for simplicity. We consider the following problem:Is any sequenceψnwhich is biorthogonal toφna generalized Riesz system?Here we put(2)CφN≔T∈Cφ;T-1∗en=ψn,n=0,1,⋯.Then we shall show in Proposition 5 that if CφN≠∅, then ψn is a generalized Riesz system and (e,T-1∗) is a constructing pair for ψn for every T∈CφN, and the mapping(3)T∈CφN↦T-1∗∈CψNis a bijection, where Cψ is the set of all constructing operators for {ψn} and(4)CψN≔K∈Cψ;K-1∗en=φn,n=0,1,⋯.Furthermore, we shall show in Proposition 6 that if there exists a bounded operator T0 in Cφ, then Cφ=T0 and CψN=T0-1∗.In Section3, we shall consider the ordered set Cφ with order ⊂ and investigate under what conditions the ordered set Cφ has a maximal element, a minimal element, the smallest element, and the largest element. First we have shown that if Dφ≔ linear span φn is dense in H, then Cφ=CφN and there exists the smallest element of Cφ, and furthermore if Dφ and D(φ)≔x∈H;∑k=0∞<x,φk>2<∞ is dense in H, there exist the smallest element of Cφ and the largest element of CψN, and in particular, if φn and ψn are regular biorthogonal sequences in H, that is, both Dφ and Dψ are dense in H, then Cφ=CφN, Cψ=CψN, and Cφ has the smallest element and the largest element. Next we shall consider the case when Dφ is not necessarily dense in H. In Theorem 14, we shall show that for a subset F of Cφ if there exists a closed operator A in H such that T⊂A for all T∈F, then F has a maximal element, and furthermore, if there exists a closed operator B in H such that T-1∗⊂B for all T∈F, then F have a maximal element and a minimal element.For the existence of the smallest element ofCφ and of the largest element of Cφ, we shall show in Theorem 16 that if there exist closed operators A and B in H such that T⊂A and T-1∗⊂B for all T∈Cφ, then Cφ has the smallest element and the largest element. Furthermore, for a biorthogonal pair (φn,ψn) of generalized Riesz systems satisfying Cφ=CφN and Cψ=CψN, we shall show in Theorem 18 that Cφ and Cψ have the smallest element and the largest element, respectively, if and only if there exist closed operators A and B in H such that T⊂A and K⊂B for all T∈Cφ and K∈Cψ. These results seem to be useful to find fitting constructing operators for each physical model because every closed operator T in H satisfying TS⊂T⊂TL belongs to Cφ, where TS is the smallest element of Cφ and TL is the largest element of Cφ.


## 2. The Basic Properties of Constructing Operators

In this section, we shall investigate the basic properties of constructing operators. Letφn be a generalized Riesz system with a constructing pair (e,T). It is easily shown that if φn is a Riesz basis, then the constructing operator T for φn is unique for e (see Proposition 1 in detail). But, in general, the constructing operators for φn are not unique, and so we put(5)Ce,φ≔T;e,Tis  a  constructing  pair  forφn.First, we investigate the relationship between Ce,φ and Cf,φ for the other ONB f=fn in H.Proposition 1.
LetT∈Ce,φ and f=fn be any ONB in H. Then the following statements hold.
(1)(f,TU∗) is a constructing pair for φn, where U is a unitary operator on H defined by Uen=fn,n=0,1,⋯, and(6)Cf,φ=TU∗;T∈Ce,φ.
(2) For the non-self-adjoint Hamiltonian, the generalized lowering operator, and the generalized raising operator forφn, we have(7)THeαT-1=TU∗HfαUT-1,TAeαT-1=TU∗AfαUT-1,TBeαT-1=TU∗BfαUT-1.Proof.
(1) This is almost trivial.
(2) This follows from(8)DHfα=UDHeα,Heα=U∗HfαU,DAfα=UDAeα,Aeα=U∗AfαU,DBfα=UDBeα,Beα=U∗BfαU.By Proposition1, we have the following.Corollary 2.
LetT∈Ce,φ and T∗=UT∗ be the polar decomposition of T∗. Then f≔U∗e is an ONB in H and T∗=TU∈Cf,φ. Furthermore, we have(9)THeαT-1=T∗HfαT∗-1,TAeαT-1=T∗AfαT∗-1,TBeαT-1=T∗BfαT∗-1.Thus we may fix an ONBe=en in H without loss of generality for investigating the properties of Ce,φ, and so throughout this paper, we fix an ONB e in H and denote Ce,φ by Cφ for simplicity. Next we consider the following problem:Suppose that(φn,ψn)is a biorthogonal pair such thatφnis a generalized Riesz system. Then, isψnalso a generalized Riesz system?LetT∈Cφ and ψnT≔T-1∗en,n=0,1,⋯. Then (φn,ψnT) is a biorthogonal pair and ψnT is a generalized Riesz system with a constructing pair (e,T-1∗). If ψnT=ψn,n=0,1⋯, then {ψn} is a generalized Riesz system with a constructing pair (e,T-1∗). But, the equality ψnT≔T-1∗en=ψn,n=0,1,⋯ does not necessarily hold. To consider when this equality holds, we define the operators Tφ,e0, Tφ,e, and Te,φ for any sequence φn in H as follows:(10)Tφ,e0≔the  linear  operator  defined  byTφ,e0en=φn,n=0,1,⋯,Tφ,e≔∑n=0∞φn⊗e¯n,Te,φ≔∑n=0∞en⊗φ¯n.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 3.
(1)Tφ,e0 and Tφ,e are densely defined linear operators in H such that(11)Tφ,e⊃Tφ,e0,Tφ,e0en=Tφ,een=φn,n=0,1,⋯.
(2)D(Te,φ)=D(φ)≔{x∈H;∑n=0∞<x,φn>2<∞} and Tφ,e0∗=Tφ,e∗=Te,φ.
(3)Tφ,e0 is closable if and only if Tφ,e is closable if and only if D(φ) is dense in H. If this holds, then(12)T¯φ,e0=T¯φ,e=Te,φ∗.From now on, let(φn,ψn) be a biorthogonal pair.Lemma 4.
Suppose thatφn is a generalized Riesz system and T∈Cφ. Then the following statements are equivalent. (i)
ψnT≔(T-1)∗en=ψn,n=0,1,⋯.(ii)
Dψ⊂D(T∗).
If this holds true, thenT is called natural.Proof.
(i)⇒(ii) This is trivial.
(ii)⇒(i) By definition of Tφ,e0, we have Tφ,e0⊂T. Furthermore, by Lemma 3, (2), we have(13)T∗⊂Tφ,e0∗=Te,φ.Take an arbitrary n∈N∪0. Then, since(14)Tφ,e0ek,ψn=φk,ψn=δkn=ek,enfor k=0,1,⋯, we have ψn∈DTφ,e0∗=D(Te,φ) and Te,φψn=en. Hence it follows from (13) that(15)T∗ψn=Te,φψn=en.Thus, we have(16)ψn=T∗-1en=ψnT.This completes the proof.We denote the set of all natural constructing operators forφn by CφN; that is,(17)CφN=T∈Cφ;ψnT≔T-1∗en=ψn,n=0,1,⋯.Then we have the following.Proposition 5.
Suppose thatφn is a generalized Riesz system. Then the following statements hold.
(1) IfCφN≠∅, then ψn is a generalized Riesz system and (e,T-1∗) is a constructing pair for ψn for every T∈CφN.
(2) Suppose thatψn is also a generalized Riesz system and put(18)Cψ=K;e,Kis  a  constructing  pair  forψn,CψN=K∈Cψ;φnK≔K-1∗en=φn,n=0,1,⋯.Then the mapping(19)T∈CφN↦T-1∗∈CψNis a bijection.
(3) Suppose thatT0∈CφN and T∈Cφ satisfying T⊂T0 or T0⊂T. Then T∈CφN. Similarly, suppose that K0∈CψN and K∈Cψ satisfying K⊂K0 or K0⊂K. Then K∈CψN.Proof.
The statements (1) and (2) are easily shown.
(3) Suppose thatT⊂T0. Then, since T0-1∗⊂T-1∗, it follows that ψn=T0-1∗en=T-1∗en=ψnT,n=0,1,⋯, which implies that T∈CφN. Similarly, we can show T∈CφN in case that T0⊂T and can show K∈CψN in case that K⊂K0 or K0⊂K. This completes the proof.As for the uniqueness of constructing operators for a generalized Riesz system we have the following.Proposition 6.
Letφn be a generalized Riesz system. Then the following statements hold.
(1) Suppose thatφn is a Riesz basis, then ψn is also a Riesz basis, Cφ=T¯φ,e=Te,ψ-1 and Cψ=T¯ψ,e=Te,φ-1.
(2) Suppose thatDφ and D(φ) are dense in H. Then we have the following.
(i) If there exists an elementT0 of Cφ such that T0 is bounded, then Cφ=T0=T¯φ,e and CψN=T0-1∗=Te,φ-1.
(ii) If there exists an elementT0 of Cφ such that T0-1 is bounded, then Cψ=T0-1∗=T¯ψ,e and CφN=T0=Te,ψ-1.Proof.
(1) Since{φn} is a Riesz basis, there exists an element T0 of Cφ such that T0 and T0-1 are bounded, which implies that T=T0=T¯φ,e=Te,ψ-1 and T-1∗=T0-1∗=T¯ψ,e=Te,φ-1 for all T∈Cφ.
(2) (i) SinceT¯φ,e⊂T0 and T0 is bounded, we have T¯φ,e=T0. Take an arbitrary T∈Cφ. Then, since T¯φ,e⊂T and T¯φ,e is bounded, we have T¯φ,e=T. Thus, Cφ=T¯φ,e. We show CψN=Te,φ-1. Take an arbitrary K∈CψN. Since K-1∗en=φn=T¯φ,een and T¯φ,e is bounded, it follows that K-1∗=T¯φ,e, which implies K=Te,φ-1. Thus, CψN=Te,φ-1.
(ii) This is similarly shown.


## 3. Ordered Structures ofCφ

In this section, we shall consider the ordered setCφ of all constructing operators for a generalized Riesz system φn with order ⊂ and investigate when Cφ has maximal elements, minimal elements, the largest element, and the smallest element. The following result gives a motivation to study the ordered structures of CφLemma 7.
Suppose thatT,S∈Cφ and T⊂S. Then, for any linear operator A such that T⊂A⊂S, the closure A¯ of A belongs to Cφ.Proof.
This is trivial.For biorthogonal sequences satisfying density-conditions, we have the following.Proposition 8.
The following statements hold.
(1) Suppose thatDφ is dense in H. Then, φn is a generalized Riesz system and Cφ=CφN, and T¯φ,e is the smallest element of Cφ. Furthermore, suppose that D(φ) is dense in H. Then, Te,φ-1 is the largest element of CψN.
(2) Suppose thatDψ is dense in H. Then, ψn is a generalized Riesz system and Cψ=CψN, and T¯ψ,e is the smallest element in Cψ. Furthermore, suppose that D(ψ) is dense in H. Then, Te,ψ-1 is the largest element in Cφ.
(3) Suppose that(φn,ψn) is regular; that is, both Dφ and Dψ are dense in H. Then, φn and ψn are generalized Riesz systems and Cφ=CφN and Cψ=CψN, and T¯φ,e is the smallest element in Cφ, T¯ψ,e is the smallest element in Cψ, Te,φ-1 is the largest element in Cψ, and Te,ψ-1 is the largest element in Cφ.Proof.
(1) We can show using Lemma3 that φn is a generalized Riesz system with a constructing pair (e,T¯φ,e) and the constructing operator T¯φ,e for φn is the smallest element in Cφ. For more detail, refer to [3]. Furthermore, a sequence ψn which is biorthogonal to φn is unique. In fact, let ψn and ψn′ be any sequences in H which are biorthogonal to φn. Then, since <ψn,φm>=δnm=<ψn′,φm> for n,m=0,1,⋯ and Dφ is dense in H, we have ψn=ψn′ for every n=0,1,⋯. We show Cφ=CφN. Take an arbitrary T∈Cφ. Then, ψnT is biorthogonal to φn. By the uniqueness of biorthogonal sequences to φn, we have ψnT=ψn,n=0,1,⋯, which implies that T∈CφN and Cφ=CφN. Suppose that Dφ and D(φ) are dense in H. We show that Te,φ-1 is the largest element in CψN. Since D(Te,φ-1)=Te,φD(Te,φ)⊃Te,φDψ=De,Te,φ-1 is a densely defined closed operator in H, and since D(Te,φ)=D(φ), it has a densely defined inverse Te,φ. Furthermore, since Te,φψn=en,n=0,1,⋯, ψn=Te,φ-1en,n=0,1,⋯. Thus we have Te,φ-1∈Cψ. Since Te,φ-1-1∗=Te,φ∗=T¯φ,e∈Cφ, it follows that Te,φ-1∈CψN. Next we show that Te,φ-1 is the largest element in CψN. Take an arbitrary K∈CψN. Then (K-1)∗∈Cφ, and so (K-1)∗en=φn, n=0,1,⋯. Hence we have Tφ,e⊂(K-1)∗. Thus K⊂Te,φ-1, and so Te,φ-1 is the largest element in CψN.
(2) This is proved at the same way as (1).
(3) SinceD(φ)⊃Dψ and D(ψ)⊃Dφ, it follows that D(φ) and D(ψ) are dense in H, which implies by (1) and (2) that the statement (3) holds.Here we give some physical examples. Letfn,n=0,1,⋯, be an ONB in L2(R) consisting of the Hermite functions which is contained in the Schwartz space S(R) of all infinitely differential rapidly decreasing functions on R. We define the moment operator p and the position operator q by(20)Dp≔the  set  of  all  differentiable  functionsfonRsuch  thatdfdx∈L2R,pfx≔-idfdx,f∈Dpand(21)Dq≔f∈L2R;∫-∞∞xfx2dx<∞,qfx≔xfx,f∈Dq.Then p and q are self-adjoint operators in L2(R) and S(R) is a core for p and q, and furthermore pS(R)⊂S(R) and qS(R)⊂S(R), and p,q≔pq-qp=-i1 on S(R). Next we define the standard bosonic operators a, a† by(22)a=12q+ip,a†=12q-ip.Then,(23)afn=0,n=0nfn-1n=1,2,⋯,a†fn=n+1fn+1,n=0,1,⋯and a,a†=1 on S(R).Example 9 (the extended quantum harmonic oscillator).
The Hamiltonian of this model is the non-self-adjoint operator, introduced in [11, 12],(24)Hβ≔β2p2+q2+i2p=βa†a+a-a†+β21,β>0.We put(25)φ0β≔e1/β2f0x-2β=e1/β2π1/4e-1/2x-2/β2.Then, φ0(β)∈S(R) and φ0(β)=e1/β2U(1/β)f0=e(a+a†)/βf0, where U(1/β) is a unitary operator defined by U(1/β)≔e(1/β)(a†-a)=e-(2/β)ip. Hence we can define a sequence φβ≔φnβ in S(R) by(26)φnβ≔1n!a†+1βnφ0β,n=1,2,⋯.Similarly, we define a sequence ψβ≔ψnβ in S(R) as follows:(27)ψ0β≔e1/β2f0x+2β,ψnβ≔1n!a†-1βnψ0β,n=1,2,⋯.Then φnβ and ψnβ are regular biorthogonal sequences in L2(R) which are generalized Riesz systems with constructing pairs (fn,e(a†+a)/β) and (fn,e-(a†+a)/β), respectively, and(28)A¯φβ≔ea†+a/βae-a†+a/β¯=a-1β,B¯φβ≔ea†+a/βa†e-a†+a/β¯=a†+1β.By Proposition 8, Tφβ,f is the smallest constructing operator and Tf,ψβ-1 is the largest constructing operator for φnβ and Tφβ,f⊂e(a†+a)/β⊂Tf,ψβ-1. Similarly, Tψβ,f is the smallest constructing operator and Tf,φβ-1 is the largest constructing operator for ψnβ and Tψβ,f⊂e-(a†+a)/β⊂Tf,φβ-1.The following example is a modification of the non-self-adjoint HamiltonianHβ in Example 9 exchanging the momentum operator p with the position operator q.Example 10.
We introduce a non-self-adjoint Hamiltonian(29)Hβ′≔β2p2+q2+2iq=βa†a+ia+a†+β2,β>0.We define sequences φβ′≔{φn′} and ψβ′≔{ψn′} in S(R) as follows:(30)φ0′≔e-i/βa†f0,φn′≔1n!a†+iβnφ0′,n=1,2,⋯and(31)ψ0′≔e1/β2ei/βa†f0,ψn′≔1n!a†-iβnψ0′,n=1,2,⋯.Then φβ′ and ψβ′ are regular biorthogonal sequences in L2(R) which are generalized Riesz systems with constructing pairs (fn,T) and (fn,T-1), respectively, where T≔e-(i/β)a†e(i/β)a¯ and T-1=e-(i/β)ae(i/β)a†¯, and TaT-1¯=a+i/β, Ta†T-1¯=a†+i/β, T-1aT¯=a-i/β, and T-1a†T¯=a†-i/β.Example 11 (the Swanson model).
The Swanson Hamiltonian, introduced in [11, 13], is a non-self-adjoint Hamiltonian(32)Hθ≔12p2+q2-i2tan⁡2θp2-q2=a†a+i2tan⁡2θa2+a†2+121,θ≠0∈-π4,π4.We define sequences φθ≔φnθ and ψθ≔ψnθ in L2(R) as follows:(33)φ0θ≔c0∑k=0∞e-itan⁡θ/2a†2f0=c0∑k=0∞-itan⁡θk2k-1!!2k!!f2k,φnθ≔1n!cos⁡θa†+isin⁡θanφ0θand(34)ψ0θ≔d0∑k=0∞eitan⁡θ/2a†2f0=d0∑k=0∞itan⁡θk2k-1!!2k!!f2k,ψnθ≔1n!cos⁡θa†-isin⁡θanψ0θ,where (2k)!!=2k(2k-2)⋯4·2, (2k-1)!!=(2k-1)(2k-3)⋯3·1, and c0 and d0 are constants satisfying <φ0(θ),ψ0(θ)>=1. Then φθ and ψθ are regular biorthogonal sequences in L2(R) contained in S(R) which are generalized Riesz systems with constructing operators Tθ≔eiθ/2a2-a†2 and Tθ-1, respectively. For the generalized lowering operator Aθ≔TθaTθ-1 and the raising operator Bθ≔Tθa†Tθ-1, we have(35)Aθ=cos⁡θa+isin⁡θa†,Bθ=cos⁡θa†+isin⁡θa.By Proposition 8, Tφθ,f(resp., Tψθ,f) is the smallest constructing operator andTψθ,f-1(resp., Tφθ,f-1) is the largest constructing operator for φθ (resp., ψθ) and every closed operatorT(resp., K) in L2(R) satisfying Tφθ,f⊂T⊂Tψθ,f-1(resp., Tψθ,f⊂K⊂Tφθ,f-1) is a constructing operator for φθ (resp., ψθ).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. Letφn be a generalized Riesz system. First we investigate under what conditions Cφ has maximal elements and minimal elements.LetC be a totally ordered subset of Cφ. Then, it is easily shown that (T-1)∗en=(S-1)∗en, n=0,1,⋯, for any T,S∈C. Hence we may put(36)ψnC=T-1∗en,T∈C,n=0,1,⋯.We have the following statements.Lemma 12.
LetC be any totally ordered subset of Cφ. The following statements hold.
(1) Suppose that∩T∈CD(T∗) is dense in H. Then there exists an upper bounded element G of C.
(2) Suppose that∩T∈CR(T) is dense in H. Then there exists a lower bounded element S of C.
(3) Suppose that∩T∈CD(T∗) and ∩T∈CR(T) are dense in H. Then for every linear operator A such that S⊂A⊂G, the closure A¯ of A belongs to Cφ.Proof.
(1) We put(37)DG=⋃T∈CDT,Gx=T0x,x∈DG,where T0 is an operator in C whose domain D(T0) contains x. Since C is totally ordered, it follows that D(G) is a subspace in H and Tx=T0x for any operators T, T0 in C whose domains contain x. Hence, G does not depend on the method of choosing T0∈C whose domain contains x. Thus G is a well-defined densely defined linear operator in H such that T⊂G for all T∈C. We show that G is closable. Indeed, we may show(38)⋂T∈CDT∗=DG∗,G∗y=T0∗y,where T0∈C whose domain D(T0) contains x. Take an arbitrary y∈∩T∈CD(T∗). Then, we have(39)Gx,y=T0x,y=x,T0∗yfor all x∈D(G), where T0∈C whose domain contains x. Hence, y∈D(G∗) and G∗y=T0∗y. Since T⊂G for all T∈C, D(G∗)⊂∩T∈CD(T∗) is trivial. Thus, (38) holds. By (38) and the assumption of (1), D(G∗) is dense in H, that is, G is closable. Next we show that G has a densely defined inverse. Suppose that Gx=0, x∈D(G). Then T0x=0 for some T0∈C, and so x=0 since T0 has an inverse. Thus G has an inverse. Since R(T)⊂R(G) and R(T) is dense in H for all T∈C, it follows that the inverse of G is densely defined, which implies that the closure G¯ of G has a densely defined closed operator in H such that G¯⊃T for all T∈C. Finally we show(40)en⊂DG¯∩DG∗-1,G¯en=φn,n=0,1,⋯.Clearly, en⊂D(T)⊂D(G¯) for all T∈C. Next we show that for any n there exists an element yn of ∩T∈CD(T∗) such that(41)en=T∗ynfor all T∈C. Indeed, take an arbitrary T∈C. Since en∈D(T∗-1)=R(T∗), there exists an element ynT of D(T∗) such that en=T∗ynT. Let any T′∈C. Since C is totally ordered, either T′⊂T or T⊂T′ holds. Suppose that T′⊂T. Since T∗⊂T′∗, it follows that ynT, ynT′∈D(T′∗) and(42)T′∗ynT=T∗ynT=en=T′∗ynT′,which implies that ynT=ynT′ since T′∗ has inverse. The equality ynT=ynT′ is similarly shown in case that T⊂T′. Hence, we have that yn≔ynT∈∩T∈CD(T∗) and T∗yn=en for all T∈C. Thus (41) holds. By (38) and (41) we have en∈R(G∗)=D(G∗-1). Furthermore, we have G¯en=Ten=φn, n=0,1,⋯ for all T∈C. Thus we have G¯∈Cφ and G¯ is an upper bounded element of C.
(2) We put(43)DS=∩T∈CDT,Sx=Tx,x∈DS,where T is any element of C. Since en⊂∩T∈CD(T) and T1x=T2x for all x∈∩T∈CD(T), S is a well-defined densely defined closed operator in H such that S⊂T for all T∈C. Hence, it is sufficient to show S∈Cφ. Since S⊂T for all T∈C and T has the inverse, S has the inverse. Furthermore, we may show(44)RS=∩T∈CRT.In fact, take an arbitrary y∈∩T∈CR(T). Since C is totally ordered, there exists an element x of D(S)=∩T∈CD(T) such that y=Tx for all T∈C. Hence, y=Sx∈R(S). The inverse inclusion R(S)⊂∩T∈CR(T) is clear. Hence (44) holds. By the assumption and (44), R(S)=D(S-1) is dense in H. Furthermore, since en⊂D(T) for all T∈C, we have en⊂D(S) and Sen=Ten=φn,n=0,1,⋯. Since S⊂T, it follows that en⊂D(T∗-1)⊂D(S∗-1). Thus, S∈Cφ and it is a lower bound of C.
(3) This follows from (1) and (2).For a subsetF of Cφ, we put(45)Fψ≔T-1∗;T∈F.Then we have the following.Lemma 13.
LetF be a subset of Cφ. Then, T0 is a maximal (resp., minimal, the largest and the smallest) element of F if and only if T0-1∗ is a minimal (resp., maximal, the smallest and the largest) element of Fψ.Proof.
Suppose thatT0 is a maximal element of F. Take an arbitrary K∈Fψ satisfying K⊂T0-1∗. Then we have that K=T-1∗ for some T∈F and T0⊂T, which implies by the maximality of T0 that T=T0 and K=T0-1∗. Thus, T0-1∗ is a minimal element of Fψ. Furthermore, we can similarly show that if T0-1∗ is a minimal element of Fψ, then T0 is a maximal element F. The other statements are similarly shown.Theorem 14.
LetF be a subset of Cφ. Then we have the following: (1)
The following statements are equivalent:
(i)
F has a maximal element.(ii)
There exists a closed operatorA in H such that T⊂A for all T∈F.(iii)
Fψ has a minimal element.(2)
The following statements are equivalent:
(i)
F has a minimal element.(ii)
There exists a closed operatorB in H such that T-1∗⊂B for all T∈F.(iii)
Fψ has a maximal element.(3)
The following statements are equivalent:
(i)
F has a maximal element and a minimal element.(ii)
There exist closed operatorsA and B in H such that T⊂A and T-1∗⊂B for all T∈F.(iii)
Fψ has a maximal element and a minimal element.Proof.
(1) (i)⇒(ii) This is trivial.
(ii)⇒(i) Suppose that there exists a closed operator A in H such that T⊂A for all T∈F. Then for any totally ordered subset C of F we have D(A∗)⊂⋂T∈CD(T∗). Hence, it follows that ⋂T∈CD(T∗) 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.
(i)⇔(iii) This follows from Lemma 13.
(2) (i)⇒(ii) This is trivial.
(ii)⇒(i) Suppose that there exists a closed operator A in H such that T-1∗⊂B for all T∈F. Then we can similarly show that Fψ has a maximal element, which implies by Lemma 13 that F has a minimal element.
(i)⇔(iii) This follows from Lemma 13.
(3) This follows from (1) and (2). This completes the proof.We remark that the closed operatorsA and B in Theorem 14 do not need any other conditions, for example, the existence of inverse.By Theorem14, we have the following.Corollary 15.
LetT0∈Cφ and put FT0=T∈Cφ;T0⊂T. Then the following statements hold.
(1) Suppose that there exists a closed operatorA in H such that T⊂A for all T∈FT0. Then there exists a maximal element of Cφ which is an extension of T0.
(2) Suppose that there exists a closed operatorB in H such that T-1∗⊂B for all T∈FT0. Then there exists a minimal element of Cφ which is a restriction of T0.Proof.
(1) By Theorem14, FT0 has a maximal element T1. Here we show that T1 is a maximal element of Cφ. Indeed, this follows since T∈FT0 for any element T of Cφ satisfying T1⊂T. We can similarly show (2).Next we investigate the existence of the smallest element and of the largest element ofCφ.Theorem 16.
C φ has the smallest element and the largest element if and only if there exist closed operators A and B in H such that T⊂A and T-1∗⊂B for all T∈Cφ.Proof.
Suppose that there exist closed operatorsA and B in H such that T⊂A and T-1∗⊂B for all T∈Cφ. We define an operator T0 as follows:(46)DT0=⋂T∈CφDT,T0x=Tx,x∈DT0,where T is an element of Cφ. Take an arbitrary x∈D(T0) and T1,T2∈Cφ. Since x∈D(T1), x∈D(T2), T1⊂A and T2⊂A, we have(47)T1x=T2x=Ax.Thus, T0 does not depend on the method of choosing T∈Cφ, and so T0 is well defined. Since en⊂∩T∈CφD(T)=D(T0), T0 is a densely defined closed operator in H such that T0⊂T for all T∈Cφ. Since T-1∗⊂B for all T∈Cφ, we have D(B∗)⊂⋂T∈CφR(T), which implies that ⋂T∈CφR(T) is dense in H. Hence, we can prove at the same way as the proof of Lemma 12 (2) that T0 is the smallest element Cφ. Next we show that Cφ has the largest element. Take an arbitrary T∈Cφ. Then ψT is a generalized Riesz system with a constructing operator T-1∗ and K⊂B and K-1∗⊂A for all K∈CψTN. Hence, as shown above there exists the smallest element K1 of CψTN, and so K1=T1-1∗ for some T1∈Cφ and T1-1∗⊂T-1∗. Thus T⊂T1 and T1 is the largest element of Cφ. The converse is trivial. This completes the proof.As seen in Section2, for a biorthogonal pair (φn,ψn), the equality T-1∗en=ψn,n=0,1,⋯ does not necessarily hold for all T∈Cφ. From this fact we define the notion of natural pair of generalized Riesz systems.Definition 17.
A biorthogonal pair(φn,ψn) of generalized Riesz systems is said to be natural, if Cφ=CφN and Cψ=CψN, that is, T-1∗en=ψn for all n and T∈Cφ and K-1∗en=φn for all n and K∈Cψ.Theorem 18.
Let(φn,ψn) be a natural pair of generalized Riesz systems. Then Cφ and Cψ have the smallest element and the largest element, respectively, if and only if there exist closed operators A and B in H such that T⊂A and K⊂B for all T∈Cφ and K∈Cψ.Proof.
This is shown using Theorem16 for the generalized Riesz systems for φn and ψn.For a generalized Riesz systemφn, suppose that there exist the largest element TL of Cφ and the smallest element TS of Cφ. Then every closed operator T in H satisfying TS⊂T⊂TL is a constructing operator of φn, and so we can construct all kinds of non-self-adjoint Hamiltonians THeαT-1, lowering operator TAeαT-1 and raising operator TBeαT-1 for φn. It may be possible to find constructing operators suitable for each of the physical models.


## 4. 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.



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