# Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases

Authors: Bilender P. Allahverdiev
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert spaceℓw2(ℤ)  (ℤ:={0,±1,±2,…}), that is, the extensions of a minimal symmetric operator with defect index (2,2)  (in the Weyl-Hamburger limit-circle cases at ±∞). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at -∞” and “dissipative at ∞.” In each case, we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also establish a functional model of the maximal dissipative operator and determine its characteristic function through the Titchmarsh-Weyl function of the self-adjoint operator. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.


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## Body

## 1. Introduction

Method of contour integration of the resolvent is one of the general methods of the spectral analysis of nonself-adjoint (dissipative) operators. It is related to a fine estimate of the resolvent on expanding contours which separates the spectrum. The feasibility of this method is restricted to weak perturbations of self-adjoint operators and operators having sparse discrete spectrum. Since there are no asymptotics of the solutions for a wide class of singular problems, this method cannot be applied properly.It is well known [1–4] that the theory of dilations with application of functional models gives an adequate approach to the spectral theory of dissipative (contractive) operators. In this theory, a key role is played by the characteristic function, which carries the full information on the spectral properties of the dissipative operator. Thus, the dissipative operator becomes the model in the incoming spectral representation of the dilation. The completeness problem of the system of eigenvectors and associated vectors is solved by the factorization of the characteristic function. The computation of the characteristic functions of dissipative operators is preceded by the construction and investigation of the self-adjoint dilation and the corresponding scattering problem, in which the characteristic function is realized as the scattering matrix [5]. The adequacy of this approach for dissipative Jacobi operators and second-order difference (or discrete Sturm-Liouville) operators has been indicated in [6–9].In this paper, we consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert spaceℓw2(ℤ), that is the extensions of a minimal symmetric operator with defect index (2,2) (in the Weyl-Hamburger limit-circle cases at ±∞). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at -∞” and “dissipative at ∞.” In each of these cases we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation according to the scheme of Lax and Phillips [5]. By means of the incoming spectral representation, we establish a functional model of the maximal dissipative operator and construct its characteristic function using the Titchmarsh-Weyl function of the self-adjoint operator. Finally, on the basis of the results obtained for the characteristic functions, we prove the theorems on completeness of the system of eigenvectors and associated vectors (or root vectors) of the maximal dissipative second-order difference operators.


## 2. Preliminaries

Lety={yn} be a sequence of complex numbers yn(n∈ℤ) and ℓ1y denote the sequence with components (ℓ1y)n. We consider the following second-order difference (or discrete Sturm-Liouville) equation on the whole line:(2.1)(l1y)n:=-an-1yn-1+bnyn-anyn+1=λwnyn,
where λ is a complex spectral parameter, wn>0,an≠0, and an,bn∈ℝ:=(-∞,∞), n∈ℤ.If we letpn=an,qn=bn-an-an-1, and Δxn=xn+1-xn, (2.1) can be written in Sturm-Liouville form as follows:(2.2)-Δ(pn-1Δyn-1)+qnyn=λwnyn,n∈Z.For arbitrary sequencesy={yn} and z={zn},n∈ℤ, we denote by [y,z] the sequence with components [y,z]n defined as:(2.3)[y,z]n:=an(ynz¯n+1-yn+1z¯n),n∈Z.
Let m,n∈ℤ with n<m. Then we have the Green’s formula:(2.4)∑j=nm[(l1y)jz¯j-yj(l1z¯)j]=[y,z]m-[y,z]n-1.For any sequencey={yn}, let ℓy denote the sequence with components (ℓy)n given by (ℓy)n=(1/wn)(ℓ1y)n, n∈ℤ. We denote by ℓw2(ℤ)(w:={wn},n∈ℤ) the Hilbert space of all complex sequences y={yn}, n∈ℤ such that ∑n=-∞∞wn|yn|2<∞, with the inner product (y,z)=∑n=-∞∞wnynz¯n. Next, we denote by D the set of all vectors y∈ℓw2(ℤ) such that ℓy∈ℓw2(ℤ). We define a maximal operatorL on D by setting Ly=ℓy.It follows from Green’s formula (2.4) that the limits [y,z]∞=limn→∞[y,z]n and [y,z]-∞=limn→-∞[y,z]n exist and are finite for arbitrary vectors y,z∈D. Therefore, taking the limit as n→-∞ and m→∞ in (2.4), for all y,z∈D, we have(2.5)(Ly,z)-(y,Lz)=[y,z]∞-[y,z]-∞.Denote byL0 the closure of the symmetric operator L0′ defined by L0′y=Ly on the linear set D0′ of finite sequences (i.e., vectors having only finitely many nonzero components) y={yn}(n∈ℤ). The minimal operatorL0 is symmetric and L0*=L. The computation of the defect index of L0 can be reduced to the computation of the defect index for the half-line case. In fact, ℓw2(ℤ) is the orthogonal sum of the space ℓw2(ℕ-), (ℕ-={-1,-2,-3,…}) and ℓw2(ℕ0) (ℕ0={0,1,2,…}) which are imbedded in the natural way in ℓw2(ℤ). Denote by L0-(L-) and L0+(L+) the minimal (maximal) operators generated by ℓ- and ℓ+ in the spaces ℓw2(ℕ-) and ℓw2(ℕ0), respectively, and D0∓(D∓) is a domain of L0∓(L∓), where (ℓ∓y)n:=(ℓy)n, n∈ℤ∖{-1,0}, (ℓ-y)-1:=(1/w-1)(-a-2y-2+b-1y-1), (ℓ+y)0:=(1/w0)(b0y0-a0y1). Then it is easy to see that the equality defL0=defL0-+defL0+ is satisfied for the defect number defL0:=dim{(L0-λI)D(L0)}⊥, Imλ≠0, of L0. This shows that the defect index of L0 has the form (k,k), where k=0,1 or 2. For defect index (0,0) the operator L0 is self-adjoint, that is, L0*=L0=L.Assume that the symmetric operatorL0 has defect index (2,2). There are several sufficient conditions that guarantee Weyl-Hamburger limit-circle cases at ±∞ (i.e., the operator L0 has defect index (2,2), see [10–17]). The domain of L0 consists of precisely those vectors y∈D satisfying the condition(2.6)[y,z]∞-[y,z]-∞=0,∀z∈D.Denote byP(1)(λ)={Pn(1)(λ)} and P(2)(λ)={Pn(2)(λ)},n∈ℤ the solutions of (2.1) satisfying the initial conditions:(2.7)P-1(1)(λ)=0,P0(1)(λ)=1,P-1(2)(λ)=-1a-1,P0(2)(λ)=0.TheWronskian of the two solutions y={yn} and z={zn},n∈ℕ of (2.1) is defined as Wn(y,z):=an(ynzn+1-yn+1zn), so that Wn(y,z)=[y,z¯]n,n∈ℤ. The Wronskian of the two solutions of (2.1) is independent of n, and the two solutions of this equation are linearly independent if and only if their Wronskian is nonzero. It follows from the conditions (2.7) and the constancy of the Wronskian that Wn(P(1),P(2))=1,n∈ℤ. Consequently, P(1)(λ) and P(2)(λ) form a fundamental system of solutions of (2.1), and P(1)(λ), P(2)(λ)∈ℓw2(ℤ) for all λ∈ℂ. The theory of difference equations can be seen in [18, 19].Letu=P(1)(0) and v=P(2)(0). Since the vectors u={un} and v={vn}(n∈ℤ) are real valued and [u,v]n=1(n∈ℤ), the following assertion can be verified easily using (2.3).Lemma 2.1.
For arbitrary vectorsy={yn}∈D and  z={zn}∈D, one has the equality:
(2.8)[y,z]n=[y,u]n[z¯,v]n-[y,v]n[z¯,u]n,(n∈Z∪{-∞,∞}).The domainD0 of the operator L0 consists of precisely those vectors y∈D satisfying the boundary conditions:(2.9)[y,u]-∞=[y,v]-∞=[y,u]∞=[y,v]∞=0.Let us consider the following linear maps fromD into ℂ2(2.10)Γ1y=([y,v]-∞[y,u]∞),Γ2y=([y,u]-∞[y,v]∞),y∈D.
Then we have the following result (see [8]).Theorem 2.2.
For any contractionK in  ℂ2 the restriction of the operator  L to the set of vectors  y∈D satisfying the boundary conditions
(2.11)(K-I)Γ1y+i(K+I)Γ2y=0,
or
(2.12)(K-I)Γ1y-i(K+I)Γ2y=0
is, respectively, a maximal dissipative or a accretive extension of the operator  L0. Conversely, every maximal dissipative (accretive) extension of  L0 is the restriction of  L to the set of vectors  y∈D satisfying (2.11) (2.12), and the contraction  K is uniquely determined by the extension. These conditions give a self-adjoint extension if and only if  K is unitary. In the latter case (2.11) and (2.12) are equivalent to the condition  (cosS)Γ1y-(sinS)Γ2y=0, where  S is a self-adjoint (Hermitian matrix) operator in  ℂ2. The general form of dissipative and accretive extensions of the operator  L0 is given by the conditions
(2.13)K(Γ1y+iΓ2y)=Γ1y-iΓ2y,Γ1y+iΓ2y∈D(K),K(Γ1y-iΓ2y)=Γ1y+iΓ2y,Γ1y-iΓ2y∈D(K),
respectively, where  K is a linear operator in  ℂ2 with  ∥Kf∥≤∥f∥,f∈D(K)⊆ℂ2. The general form of symmetric extensions is given by the formulae (2.13), where  K is an isometric operator.
In particular, ifK is a diagonal matrix, the boundary conditions
(2.14)[y,v]-∞-h1[y,u]-∞=0,(2.15)[y,u]∞-h2[y,v]∞=0,
with  Imh1≥0 or  h1=∞, and  Imh2≥0 or  h2=∞(Imh1≤0 or h1=∞, and Imh2≤0 or h2=∞) describe all the maximal dissipative (maximal accretive) extensions of  L0 with separated boundary conditions. The self-adjoint extensions of  L0 are obtained precisely when  Imh1=0 or  h1=∞, and  Imh2=0 or  h2=∞. Here for  h1=∞(h2=∞)  condition (2.14) (2.15) should be replaced by  [y,u]-∞=0([y,v]∞=0).In what follows, we will study the dissipative operatorsLh1h2∓ generated by ℓ and the boundary conditions (2.14) and (2.15) of two types: “dissipative at -∞,” that is, when either Imh1>0 and Imh2=0  orh2=∞; “dissipative at ∞,” when Imh1=0  orh1=∞ and Imh2>0.


## 3. Self-Adjoint Dilations of the Maximal Dissipative Operators

In order to construct a self-adjoint dilation of the maximal dissipative operatorLh1h2- in the case of “dissipative at -∞” (i.e., Imh1>0 and Imh2=0 or h2=∞), we associate with H:=ℓw2(ℤ) the “incoming” and “outgoing” channels D-:=L2(-∞,0) and D+:=L2(0,∞), we form the orthogonal sum ℋ=D-⊕H⊕D+ and we call it the main Hilbert space of the dilation. In the space ℋ, we consider the operator ℒh1h2- generated by the expression(3.1)L〈φ-,y,φ+〉=〈idφ-dξ,ly,idφ+dς〉
on the set D(ℒh1h2-) of vectors 〈φ-,y,φ+〉 satisfying the conditions φ-∈W21(-∞,0), φ+∈W21(0,∞), y∈D and(3.2)[y,v]-∞-h1[y,u]-∞=αφ-(0),[y,v]-∞-h¯1[y,u]-∞=αφ+(0),[y,u]∞-h2[y,v]∞=0,
where W21 denotes the Sobolev space and α2:=2Imh1, α>0.Theorem 3.1.
The operatorℒh1h2- is self-adjoint in  ℋ and it is a self-adjoint dilation of the maximal dissipative operator  Lh1h2-.Proof.
We assume thatf,g∈D(ℒh1h2-) with f=〈φ-,y,φ+〉 and g=〈ψ-,z,ψ+〉. Then using integration by parts and (3.1), we obtain
(3.3)(Lh1h2-f,g)H=∫0∞iφ-′ψ¯-dξ+(Ly,z)H+∫0∞iφ+′ψ¯+dξ=iφ-(0)ψ¯-(0)-iφ+(0)ψ¯+(0)+[y,z]∞-[y,z]-∞+(f,Lh1h2-g)H.
If we use the boundary conditions (3.2) for the components of the vectors f,g and Lemma 2.1, we see that iφ-(0)ψ¯-(0)-iφ+(0)ψ¯+(0)+[y,z]∞-[y,z]-∞=0. Thus, ℒh1h2- is symmetric. Therefore, to prove that ℒh1h2- is self-adjoint, it is sufficient to show that (ℒh1h2-)*⊆ℒh1h2-. Let us take g=〈ψ-,z,ψ+〉∈D((ℒh1h2-)*) and let (ℒh1h2-)*g=g*=〈ψ-*,z*,ψ+*〉∈ℋ so that
(3.4)(Lh1h2-f,g)H=(f,g*)H,∀f∈D(Lh1h2-).
If we choose the components of f∈D(ℒh1h2-) properly in (3.4), it becomes easy to show that ψ-∈W21(-∞,0), ψ+∈W21(0,∞), z∈D, and g*=ℒg, where the operator ℒ is given by (3.1). As a result, (3.4) takes the form (ℒf,g)ℋ=(f,ℒg)ℋ, for all f∈D(ℒh1h2-). Hence, in the bilinear form (ℒf,g)ℋ, the sum of the integral terms must be equal to zero:
(3.5)iφ-(0)ψ¯-(0)-iφ+(0)ψ¯+(0)+[y,z]∞-[y,z]-∞=0,
for all f=〈φ-,y,φ+〉∈D(ℒh1h2-). In addition, if we solve the boundary conditions (3.2) for [y,u]-∞ and [y,v]-∞, we get
(3.6)[y,u]-∞=-iα(φ+(0)-φ-(0)),[y,v]-∞=αφ-(0)-ih1α(φ+(0)-φ-(0)).
It follows from Lemma 2.1 and (3.6) that (3.5) is equivalent to the following equality:
(3.7)iφ-(0)ψ¯-(0)-iφ+(0)ψ¯+(0)=[y,z]-∞-[y,z]∞=-iα(φ+(0)-φ-(0))[z¯,v]-∞-α[φ-(0)-ih1α2(φ+(0)-φ-(0))][z¯,u]-∞-[y,u]∞[z¯,v]∞+[y,v]∞[z¯,u]∞=-iα(φ+(0)-φ-(0))[z¯,v]-∞-α[φ-(0)-ih1α2(φ+(0)-φ-(0))][z¯,u]-∞+([z¯,u]∞-h2[z¯,v]∞)[y,v]∞.
Note that the values φ±(0) can be any complex numbers. Therefore, when we compare the coefficients of φ±(0) on the left and right of the last equality we see that the vector g=〈ψ-,z,ψ+〉 satisfies the boundary conditions [z,v]-∞-h1[z,u]-∞=αψ-(0), [z,v]-∞-h¯1[z,u]-∞=αψ+(0), [z,u]∞-h2[z,v]∞=0. Consequently, we obtain (ℒh1h2-)*⊆ℒh1h2-, and hence ℒh1h2-=(ℒh1h2-)*.
The self-adjoint operatorℒh1h2- generates in ℋ a unitary group Ut-=exp[iℒh1h2-t],  t∈ℝ. Let P:ℋ→H and P1:H→ℋ denote the mappings acting according to the formulas P:〈φ-,y,φ+〉→y and P1:y→〈0,y,0〉. Let Zt-=PUt-P1t≥0. The family {Zt-}, t≥0, of operators is a strongly continuous semigroup of completely nonunitary contractions on H. (We recall that the linear bounded operator A acting in the Hilbert space H is calledcompletely nonunitaryif invariant subspace M⊆H (M≠{0} of operator A whose restriction to M is unitary, does not exist). Let us denote by Ah1h2 the generator of this semigroup: Ah1h2y=limt→+0(it)-1(Zt-y-y). The domain of Ah1h2 consists of all the vectors for which the limit exists. Ah1h2 is a maximal dissipative operator. The operator ℒh1h2- is called the self-adjoint dilation of Ah1h2 [1–4]. We show that Ah1h2=Lh1h2-, and thus ℒh1h2- is a self-adjoint dilation of Lh1h2-. To do this, we first verify the equality [1–4]:
(3.8)P(Lh1h2--λI)-1P1y=(Lh1h2--λI)-1y,y∈H,Im⁡λ<0.
Denote (ℒh1h2--λI)-1P1y=g=〈ψ-,z,ψ+〉. Then (ℒh1h2--λI)g=P1y, and hence Lz-λz=y, ψ-(ξ)=ψ-(0)e-iλξ and ψ+(ς)=ψ+(0)e-iλς. Since g∈D(ℒh1h2-), and hence, ψ-∈L2(-∞,0); it follows that ψ-(0)=0, and, consequently, z satisfies the boundary conditions [z,v]-∞-h1[z,u]-∞=0, [z,u]∞-h2[z,v]∞=0. Therefore, z∈D(Lh1h2-) and since a point λ with Imλ<0 cannot be an eigenvalue of a dissipative operator, it follows that z=(Lh1h2--λI)-1y. Note that ψ+(0) is obtained from the formula ψ+(0)=α-1([z,v]-∞-h¯1[z,u]-∞). Then
(3.9)(Lh1h2--λI)-1P1y=〈0,(Lh1h2--λI)-1y,α-1([z,v]-∞-h¯1[z,u]-∞)e-iλς〉,
for y∈H and Imλ<0. By applying P, one can obtain (3.8).
Now, it is not difficult to show thatAh=Lh-. In fact, it follows from (3.8) that
(3.10)(Lh1h2--λI)-1=P(Lh1h2--λI)-1P1=-iP∫0∞Ut-e-iλtdtP1=-i∫0∞Zt-e-iλtdt=(Ah1h2-λI)-1,Im⁡λ<0,
and thus Lh1h2-=Ah1h2. Theorem 3.1. is proved.In order to construct a self-adjoint dilation of the maximal dissipative operatorLh1h2+ in the case “dissipative at ∞” (i.e., Imh1=0 or h1=∞ and Imh2>0) in ℋ, we consider the operator ℒh1h2+ generated by the expression (3.1) on the set D(ℒh1h2+) of vectors 〈φ-,y,φ+〉 satisfying the conditions φ-∈W21(-∞,0), φ+∈W21(0,∞), y∈D and(3.11)[y,v]-∞-h1[y,u]-∞=0,[y,u]∞-h2[y,v]∞=αφ-(0),[y,u]∞-h¯2[y,v]∞=αφ+(0),
where α2:=2Imh2,α>0.The proof of the next theorem is similar to that of Theorem3.1.Theorem 3.2.
The operatorℒh1h2+ is self-adjoint in  ℋ and it is a self-adjoint dilation on the maximal dissipative operator  Lh1h2+.


## 4. Scattering Theory of the Dilations and Functional Models of the Maximal Dissipative Operators

The unitary groupUt±=exp[iℒh1h2±t] (t∈ℝ) has a crucial property which enables us to apply the Lax-Phillips scheme [5]. In other words, it has incoming and outgoing subspaces D-=〈L2(-∞,0),0,0〉 and D+=〈0,0,L2(0,∞)〉 satisfying the following properties:(1)
Ut±D-⊂D-, t≤0 and Ut±D+⊂D+, t≥0;(2)
⋂t≤0Ut±D-=⋂t≥0Ut±D+={0};(3)
⋃t≥0Ut±D-¯=⋃t≤0Ut±D+¯=ℋ;(4)
D-⊥D+.Property (4) is obvious. To verify property (1) forD+ (the proof for D- is similar), we set Rλ±=(ℒh1h2±-λI)-1, for all λ with Imλ<0. Then, for any f=〈0,0,φ+〉∈D+, we have(4.1)Rλ±f=〈0,0,-ie-iλς∫0ςe-iλsφ+(s)ds〉.
Hence, we find Rλf∈D+. Therefore, if g⊥D+, then it follows that(4.2)0=(Rλ±f,g)H=-i∫0∞e-iλt(Ut±f,g)Hdt,Im⁡λ<0.From this, we conclude that(Ut±f,g)ℋ=0 for all t≥0. Hence Ut±D+⊂D+, for t≥0, which completes the proof of property (1).To prove property (2), we denote byP+:ℋ→L2(0,∞) and P1+:L2(0,∞)→D+ the mappings acting according to the formulas P+:〈φ-,u,φ+〉→φ+ and P1+:φ→〈0,0,φ〉, respectively. Note that the semigroup of isometries Vt±=P+Ut±P1+, t≥0 is a one-sided shift in L2(0,∞). Indeed, the generator of the semigroup of the one-sided shift Vt in L2(0,∞) is the differential operator i(d/dξ) satisfying the boundary condition φ(0)=0. On the other hand, the generator A± of the semigroup of isometries Vt±, t≥0, is the operator A±φ=P+ℒh1h2±P1+f=P+ℒh1h2±〈0,0,φ〉=P+〈0,0,i(dφ/dξ)〉=i(dφ/dξ), where φ∈W21(0,∞) and φ(0)=0. As a semigroup is uniquely determined by its generator, it follows that Vt±=Vt, and thus, ⋂t≥0Ut±D+=〈0,0,⋂t≥0VtL2(0,∞)〉={0}, which verifies the property (2).The scattering matrix is defined in terms of the spectral representations theory in this scheme of the Lax-Phillips scattering theory. We will continue with their construction and prove property (3) of the incoming and outgoing subspaces along the way.We recall that the linear operatorA (with domain D(A)) acting in the Hilbert space H is calledcompletely nonself-adjoint (orsimple)if the invariant subspace M⊆D(A) (M≠{0}) of the operator A whose restriction to M is self-adjoint, does not exist.Lemma 4.1.
The operatorLh1h2±  is completely nonself-adjoint (simple).Proof.
LetH′⊂H be a nontrivial subspace where Lh1h2- (the proof for Lh1h2+ is similar) induces a self-adjoint operator L′ with domain D(L′)=H′∩D(Lh1h2-). If f∈D(L′), then we get f∈D(L′*) and [y,u]-∞-h1[y,v]-∞=0, [y,u]-∞-h¯1[y,v]-∞=0. It follows that [y,u]-∞=0, [y,v]-∞=0 and y(λ)=0 for the eigenvectors y(λ) of the operator Lh1h2- that lie in H′ and are eigenvectors of L′. Since all solutions of (2.1) belong to ℓw2(ℤ), we conclude that the resolvent Rλ(Lh1h2-) of the operator Lh1h2- is a Hilbert-Schmidt operator, and hence the spectrum of Lh1h2- is purely discrete. Using the theorem on expansion in eigenvectors of the self-adjoint operator L′, we see that H′={0}, that is, the operator Lh1h2- is simple. The lemma is proved.To prove property (3) we first set(4.3)H-±=⋃t≥0Ut±D-¯,H+±=⋃t≤0Ut±D+¯,
and prove the following lemma.Lemma 4.2.
The equalityℋ-±+ℋ+±=ℋ holds.Proof.
Using property (1) of the subspaceD±, we can easily show that the subspace ℋ±′=ℋ⊖(ℋ-±+ℋ+±) is invariant with respect to the group {Ut±} and has the form ℋ±′=〈0,H±′,0〉, where H±′ is a subspace in H. Accordingly, if the subspace ℋ±′ (and thus, H±′ as well) were nontrivial, then the unitary group {Ut±′}, restricted to this subspace, would be a unitary part of the group {Ut±}, and thus the restriction Lh1h2±′ of Lh1h2± to H±′ would be a self-adjoint operator in H±′. It follows from the simplicity of the operator Lh1h2±that H±′={0}, that is, ℋ±′={0}. The proof is completed.Letφ(λ) and ψ(λ) be the solutions of (2.1) satisfying the conditions:(4.4)[φ,u]-∞=-1,[φ,v]-∞=0,[ψ,u]-∞=0,[ψ,v]-∞=1.TheTitchmarsh-Weyl functionm∞h2(λ) of the self-adjoint operator L∞h2- is determined by the condition [ψ+m∞h2φ,u]∞-h2[ψ+m∞h2φ,v]∞=0. Then, we have(4.5)m∞h2(λ)=-[ψ,u]∞-h2[ψ,v]∞[φ,u]∞-h2[φ,v]∞.
The last equality implies that m∞h2(λ) is a meromorphic function on the complex plane ℂ with a countable number of poles on the real axis, which coincide with the eigenvalues of the self-adjoint operator L∞h2. One can also show that the function m∞h2(λ) has the following properties: ImλImm∞h2(λ)>0 for Imλ≠0 and m∞h2(λ¯)=m∞h2(λ)¯ for complex λ with the exception of the real poles of m∞h2(λ).We adopt the following notations:θ(λ)=ψ(λ)+m∞h2(λ)φ(λ),(4.6)Sh1h2-(λ)=m∞h2(λ)-h1m∞h2(λ)-h¯1.Let(4.7)Uλ-(ξ,ς)=〈e-iλξ,(m∞h2(λ)-h1)-1αθ(λ),S̅h1h2-(λ)e-iλς〉.
For real values of λ, the vectors Uλ-(ξ,ς) do not belong to the space ℋ, but they satisfy the equation ℒU=λU and the boundary conditions (3.2). Using Uλ-(ξ,ς), we define the transformation F-:f→f̃-(λ) by (F-f)(λ):=f̃-(λ):=(1/2π)(f,Uλ-)ℋ on the vector f=〈φ-,y,φ+〉, where φ-, φ+ are smooth, compactly supported functions, and y={yn},n∈ℤ, is a finite sequence.Lemma 4.3.
The transformationF- isometrically maps  ℋ- onto  L2(ℝ). For all vectors  f,g∈ℋ--, the Parseval equality and the inversion formula hold:
(4.8)(f,g)H=(f̃-,g̃-)L2=∫-∞∞f̃-(λ)g̃-(λ)¯dλ,f=12π∫-∞∞f̃-(λ)Uλ-dλ,
where  f̃-(λ)=(F-f)(λ) and  g̃-(λ)=(F-g)(λ).Proof.
Forf,g∈D-, f=〈φ-,0,0〉, g=〈ψ-,0,0〉, we have
(4.9)f̃-(λ):=12π(f,Uλ-)H=12π∫-∞0φ-(ξ)eiλξdξ∈H-2,
and, by the usual Parseval equality for Fourier integrals,
(4.10)(f,g)H=∫-∞0φ-(ξ)ψ-(ξ)¯dξ=∫-∞∞f̃-(λ)g̃-(λ)¯dλ=(F-f,F-g)L2.
From now on, let H±2 denote the Hardy classes in L2(ℝ) consisting of the functions which are analytically extendable to the upper and lower half-planes, respectively.
Let us extend the Parseval equality to the wholeℋ--. To this end, we consider in ℋ-- the dense set ℋ-′ of vectors obtained from the smooth, compactly supported functions in D-:f∈ℋ-′ if f=UT-f0, f0=〈φ-,0,0〉, φ-∈C0∞(-∞,0), where T=Tf is a nonnegative number (depending on f). In this case, if f,g∈ℋ-′, then U-T-f,U-T-g∈D- for T>Tf and T>Tg. Furthermore, the first components of these vectors belong to C0∞(-∞,0). Since the operators Ut-,t∈ℝ, are unitary, the equality F-U-T-f=(U-T-f,Uλ-)ℋ=e-iλT(f,Uλ-)ℋ=e-iλTF-f gives us that
(4.11)(f,g)H=(U-T-f,U-T-g)H=(F-U-T-f,F-U-T-g)L2=(e-iλTF-f,e-iλTF-g)L2=(F-f,F-g)L2.
If we take the closure in (4.11), we get the Parseval equality for the whole space ℋ--. The inversion formula follows from the Parseval equality if all integrals in it are considered as limits in the mean of integrals over finite intervals. In conclusion, we have F-ℋ--=⋃t≥0F-Ut-D-¯=⋃t≥0e-iλtH-2¯=L2(ℝ) which implies that F- maps ℋ-- onto the whole of L2(ℝ). The lemma is proved.Now, we let(4.12)Uλ+(ξ,ς)=〈Sh1h2-(λ)e-iλξ,(m∞h2-(λ)-h¯1)-1αθ(λ),e-iλς〉.
Note as in the previous case that the vectors Uλ+(ξ,ς), for real values of λ, do not belong to the space ℋ. But, Uλ+(ξ,ς) satisfies the equation ℒU=λU, λ∈ℝ, and the boundary conditions (3.2). By means of Uλ+(ξ,ς), we consider the transformation F+:f→f̃+(λ) by setting (F+f)(λ):=f̃+(λ):=(1/2π)(f,Uλ+)ℋ on vectors f=〈φ-,y,φ+〉, where φ-, φ+ are smooth, compactly supported functions, and y={yn}, n∈ℤ, is a finite sequence. The proof of the next result is similar to that of Lemma 4.3.Lemma 4.4.
The transformationF+ isometrically maps  ℋ+- onto  L2(ℝ), and for all vectors  f,g∈ℋ+-, the Parseval equality and the inversion formula hold:
(4.13)(f,g)H=(f̃+,g̃+)L2=∫-∞∞f̃+(λ)g̃+(λ)¯dλ,f=12π∫-∞∞f̃+(λ)Uλ+dλ,
where  f̃+(λ)=(F+f)(λ) and  g̃+(λ)=(F+g)(λ).From (4.6), we see that |Sh1h2-(λ)|=1 for all λ∈ℝ. Therefore, it follows from the explicit formula for the vectors Uλ+ and Uλ- that(4.14)Uλ-=S¯h1h2-(λ)Uλ+,(λ∈R).
Lemmas 4.3 and 4.4 imply that ℋ--=ℋ+-. Together with Lemma 4.2, this results in ℋ=ℋ--=ℋ+- and the property (3) of the incoming and outgoing subspaces for Ut-.Therefore, the transformationF- maps isometrically onto L2(ℝ) with the subspace D- mapped onto H-2 and the operators Ut- are transformed into the operators of multiplication by eiλt, that is, F- is the incoming spectral representation for the group {Ut-}. Similarly F+ is the outgoing spectral representation for {Ut-}. It is seen from (4.14) that we can realize the passage from the F+-representation of a vector f∈ℋ to its F--representation multiplying by the function Sh1h2-(λ):f̃-(λ)=Sh1h2-(λ)f̃+(λ). According to [5], the scattering function (matrix) of the group {Ut-} with respect to the subspaces D- and D+, is the coefficient by which the F--representation of a vector f∈ℋ must be multiplied in order to get the corresponding F+-representation: f̃+(λ)=S¯h1h2-(λ)f̃-(λ) and, thus, we have proved the following theorem.Theorem 4.5.
The functionS¯h1h2-(λ) is the scattering matrix of the group  {Ut-} (of the self-adjoint operator  ℒh1h2-).LetS(λ) be an arbitrary nonconstant inner function [1–4] on the upper half-plane (the analytic function S(λ) on the upper half-plane ℂ+ is called inner function on ℂ+ if |S(λ)|≤1 for λ∈ℂ+ and |S(λ)|=1 for almost all λ∈ℝ). Let 𝒦=H+2⊖SH+2. We can see that 𝒦≠{0} is a subspace of the Hilbert space H+2. Now, let us consider the semigroup of the operators Zt, t≥0, acting in 𝒦 according to the formula Ztφ=P[eiλtφ], φ:=φ(λ)∈𝒦, where P denotes the orthogonal projection from H+2 onto 𝒦. The generator of the semigroup {Zt} is denoted by T:Tφ=limt→+0(it)-1(Ztφ-φ), which is a maximal dissipative operator acting in 𝒦 with the domain D(T) consisting of all vectors φ∈𝒦, so that the limit exists. The operator T is called a model dissipative operator (note that this model dissipative operator, which is associated with the names of Lax and Phillips [5], is a special case of a more general model dissipative operator constructed by Sz-Nagy and Foiaş [1, 2]). The basic assertion is that S(λ) is the characteristic function of the operator T.LetK=〈0,H,0〉 so that ℋ=D-⊕K⊕D+. It can be concluded from the explicit form of the unitary transformation F-  that(4.15)H⟶L2(R),f⟶f̃-(λ)=(F-f)(λ),D-⟶H-2,D+⟶Sh1h2-H+2,K⟶H+2⊖Sh1h2-H+2,Ut-f⟶(F-Ut-F--1f̃-)(λ)=eiλtf̃-(λ).
The formulas (4.15) show that the operator Lh1h2- is unitarily equivalent to the model dissipative operator with the characteristic function Sh1h2-(λ). Since the characteristic functions of unitarily equivalent dissipative operators coincide [1–4], we have proved the theorem below.Theorem 4.6.
The characteristic function of the maximal dissipative operatorLh1h2- coincides with the function  Sh1h2-(λ) defined in (4.6).Ifmh1∞(λ) is the Titchmarsh-Weyl function of the self-adjoint operator Lh1∞, then it can be expressed in terms of the Wronskian of the solutions as follows:(4.16)mh1∞(λ)=-[χ,v]∞[ϕ,v]∞.
Here ϕ(λ) and χ(λ) are solutions of (2.1) and normalized by(4.17)[ϕ,u]-∞=-11+h12,[ϕ,v]-∞=-h11+h12,[χ,u]-∞=h11+h12,[χ,v]-∞=11+h12.
Let us adopt the following notations:(4.18)n(λ):=[ϕ,u]∞[χ,v]∞,m(λ):=mh1∞(λ),S+(λ):=Sh1h2+(λ):=m(λ)n(λ)-h2m(λ)n(λ)-h¯2.Let(4.19)Vλ-(ξ,ς)=〈e-iλξ,αm(λ)[(m(λ)n(λ)-h2)[χ,v]∞]-1ϕ(λ),S¯+(λ)e-iλς〉.
One can see that the vector Vλ-(ξ,ς) does not belong to ℋ for λ∈ℝ, but Vλ- satisfies the equation ℒV=λV, λ∈ℝ, and the boundary conditions (3.11). By means of Vλ-, we define the transformation F-:f→f̃-(λ) by (F-f)(λ):=f̃-(λ):=(1/2π)(f,Vλ-)ℋ on the vector f=〈φ-,y,φ+〉, where φ-, φ+ are smooth, compactly supported functions, and y={yn},n∈ℤ, is a finite sequence. The next result can be proved following the steps similar to the proof of Lemma 4.3.Lemma 4.7.
The transformationF- isometrically maps  ℋ-+ onto  L2(ℝ). For all vectors  f,g∈ℋ-+, the Parseval equality and the inversion formula hold:
(4.20)(f,g)H=(f̃-,g̃-)L2=∫-∞∞f̃-(λ)g̃-(λ)¯dλ,f=12π∫-∞∞f̃-(λ)Uλ-dλ,
where  f̃-(λ)=(F-f)(λ) and  g̃-(λ)=(F-g)(λ).Let(4.21)Vλ+(ξ,ς)=〈S+(λ)e-iλξ,αm(λ)[(m(λ)n(λ)-h¯2)[χ,v]∞]-1ϕ(λ),e-iλς〉.
The vector Vλ+(ξ,ς) does not belong to ℋ for λ∈ℝ. However, Vλ+ satisfies the equation ℒV=λV, λ∈ℝ, and the boundary conditions (3.11). Using Vλ+(ξ,ς), let us consider the transformation F+:f→f̃+(λ) on vectors f=〈φ-,y,φ+〉, in which φ-,φ+ are smooth, compactly supported functions, and y={yn},n∈ℤ, is a finite sequence, by setting (F+f)(λ):=f̃+(λ):=(1/2π)(f,Uλ+)ℋ.Lemma 4.8.
The transformationF+ isometrically maps  ℋ++ onto  L2(ℝ). For all vectors  f,g∈ℋ++, the Parseval equality and the inversion formula hold:
(4.22)(f,g)H=(f̃+,g̃+)L2=∫-∞∞f̃-(λ)g̃-(λ)¯dλ,f=12π∫-∞∞f̃+(λ)Uλ+dλ,
where  f̃+(λ)=(F+f)(λ) and  g̃+(λ)=(F+g)(λ).It is seen from (4.18) that the function Sh1h2+(λ) satisfies |Sh1h2+(λ)|=1 for λ∈ℝ. Therefore, the explicit formula for the vectors Uλ+ and Uλ- gives us that(4.23)Vλ-=S¯h1h2+(λ)Vλ+,λ∈R.
Hence, we conclude the equality ℋ-+=ℋ++ from Lemmas 4.7 and 4.8. Together with Lemma 4.2, we get ℋ=ℋ-+=ℋ++. We can see from (4.23) that the passage from the F--representation of a vector f∈ℋ to its F+-representation is realized as follows: f̃+(λ)=S¯h1h2+(λ)f̃-(λ). Thus, we have proved the following assertion.Theorem 4.9.
The functionS¯h1h2+(λ) is the scattering matrix of the group  {Ut+} (of the self-adjoint operator  ℒh1h2+).Using the explicit form of the unitary transformationF-, we obtain(4.24)H⟶L2(R),f⟶f̃-(λ)=(F-f)(λ),D-⟶H-2,D+⟶Sh1h2+H+2,K⟶H+2⊖Sh1h2+H+2,Ut+f⟶(F-Ut+F--1f̃-)(λ)=eiλtf̃-(λ).
We conclude from (4.24) that the operator Lh1h2+ is a unitary equivalent to the model dissipative operator with characteristic function Sh1h2+(λ), which in turn proves the next theorem.Theorem 4.10.
The characteristic function of the maximal dissipative operatorLh1h2+ coincides with the function  Sh1h2+(λ) defined by (4.18).


## 5. Completeness Theorems for the System of Eigenvectors and Associated Vectors of the Maximal Dissipative Operators

We know that the characteristic function of a maximal dissipative operatorLh1h2± carries complete information about the spectral properties of this operator [1–4]. For example, completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators Lh1h2± is guaranteed by the absence of a singular factor of the characteristic function Sh1h2±(λ) in the factorization Sh1h2±(λ)=S±(λ)B±(λ) (where B±(λ) is a Blaschke product).LetA be a linear operator in the Hilbert space H with the domain D(A). The complex number λ0 is called an eigenvalue of the operator A if there exists a nonzero element y0∈D(A) satisfying Ay0=λ0y0. Such an element y0 is called the eigenvector of the operator A corresponding to the eigenvalue λ0. The elements y1,y2,…,yk are called the associated vectors of the eigenvector y0 if they belong to D(A) and satisfy Ayj=λ0yj+yj-1, j=1,2,…,k. The element y∈D(A),y≠0 is called a root vector of the operator A corresponding to the eigenvalue λ0, if all powers of A are defined on this element and (A-λ0I)my=0 for some integer m. The set of all root vectors of A corresponding to the same eigenvalue λ0 with the vector y=0 forms a linear set Nλ0 and is called the root lineal. The dimension of the lineal Nλ0 is called the algebraic multiplicity of the eigenvalue λ0. The root lineal Nλ0 coincides with the linear span of all eigenvectors and associated vectors of A corresponding to the eigenvalue λ0. Therefore, the completeness of the system of all eigenvectors and associated vectors of A is equivalent to the completeness of the system of all root vectors of this operator.Theorem 5.1.
For all values ofh1 with Imh1>0, except possibly for a single value  h1=h10, and for fixed  h2(Imh2=0 or  h2=0), the characteristic function  Sh1h2-(λ) of the maximal dissipative operator  Lh1h2- is a Blaschke product and the spectrum of  Lh1h2- is purely discrete and belongs to the open upper half plane. The operator  Lh1h2-(h1≠h10) has a countable number of isolated eigenvalues with finite algebraic multiplicity and limit points at infinity, and the system of all eigenvectors and associated vectors (or root vectors) of this operator is complete in the space  ℓw2(ℤ).Proof.
It can be easily seen from (4.6) that Sh1h2-(λ) is an inner function in the upper half-plane and, moreover, it is meromorphic in the whole λ-plane. Then, it can be factorized as
(5.1)Sh1h2-(λ)=eiλcBh1h2(λ),c=c(h1)≥0,
where Bh1h2(λ) is a Blaschke product. It can be inferred from (5.1) that
(5.2)|Sh1h2-(λ)|≤e-c(h1)Im⁡λ,Im⁡λ≥0.
Further, if we express m∞h2(λ) in terms of Sh1h2-(λ) and then use (4.6), we find
(5.3)m∞h2(λ)=h¯1Sh1h2-(λ)-h1Sh1h2-(λ)-1.
Ifc(h1)>0 for a given value h1 (Imh1>0), then limt→+∞Sh1h2-(it)=0 follows from (5.2). Hence, we obtain limt→+∞m∞h2(it)=h1 in the light of (5.3). Since m∞h2(λ) is independent of h1, c(h1) can be nonzero at not more than a single point h1=h10 (and, further, h10=limt→+∞m∞h2(it)). Hence, the theorem is proved.The proof of the next result is similar to that of Theorem5.1.Theorem 5.2.
For all values ofh2 with  Imh2>0, except possibly for a single value  h2=h20, and for fixed  h1(Imh1=0 or  h1=∞), the characteristic function  Sh1h2+(λ) of the maximal dissipative operator  Lh1h2+ is a Blaschke product and the spectrum of  Lh1h2+ is purely discrete and belongs to the open upper half-plane. The operator  Lh1h2+(h2≠h20) has a countable number of isolated eigenvalues with finite algebraic multiplicity and limit points at infinity, and the system of all eigenvectors and associated vectors of this operator is complete in the space  ℓw2(ℤ).Since a linear operatorS acting in a Hilbert space H is maximal accretive if and only if -S is maximal dissipative, all results obtained for maximal dissipative operators can be immediately transferred to maximal accretive operators.



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*Source: 473461-2012-06-10.xml*