ISRN.MATHEMATICAL.ANALYSIS ISRN Mathematical Analysis 2090-4665 Hindawi Publishing Corporation 471640 10.1155/2014/471640 471640 Research Article Self-Adjoint Extension and Spectral Theory of a Linear Relation in a Hilbert Space http://orcid.org/0000-0003-3551-7141 Acharya Keshav Raj Franchi B. Karakostas G. L. Department of Mathematics, Southern Polytechnic State University, Marietta, GA 30060 USA spsu.edu 2014 532014 2014 21 12 2013 28 01 2014 5 3 2014 2014 Copyright © 2014 Keshav Raj Acharya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The aim of this paper is to develop the conditions for a symmetric relation in a Hilbert space to have self-adjoint extensions in terms of defect indices and discuss some spectral theory of such linear relation.

1. Introduction

In this paper, we discuss the theory of linear relations in a Hilbert space. These linear relations were first studied by Arnes, Coddington, Dijksma, de Snoo, and Hassi et al in . It has also been studied extensively more recently in . The theory has particular interest because, in some of the application problems, a linear operator can have multivalued part; for example, see [6, 7]. Here, we concentrate on establishing the conditions for symmetric relations to have self-adjoint extensions in terms of defect indices. Moreover, we discuss the spectral theory of such self-adjoint relations. The analogous treatment on operator theory of some of the theorems on this paper can be found in .

Let be a Hilbert space over and denote by 2 the Hilbert space . A linear relation ={(f,g):f,g} on is a subspace of 2. The graph of an operator is an example of a linear relation but note that a relation can have multivalued part.

These relations have been used in some of the eigenvalue problems in ordinary differential equations. For example, the canonical systems (1)Ju(x)=zH(x)u(x),z, where J=(0-110) and H(x) is a 2×2 positive semidefinite matrix whose entries are locally integrable, induce a multivalued linear relation. For instance, we may think of writing the systems in the form (2)H(x)-1Ju=zu and consider it as an operator on a Hilbert space. But H(x) is not invertible in general therefore can not be considered as an eigenvalue equation of an operator. Instead, the system induces a linear relation that may have a multivalued part. This is one of the main motivations for our work in this paper. The boundary value problem of such canonical systems has been studied by using linear relations; see [6, 7, 9].

D ( ) = { f : ( f , g ) } and R()={g:(f,g)} are respectively defined as the domain and range of the relation .

- 1 = { ( g , f ) : ( f , g ) } denotes the inverse relation. The adjoint of on is a closed linear relation defined by (3)*={(h,k)2:g,h=f,k,(f,g)}. A linear relation 𝒮 is called symmetric if 𝒮𝒮* and self-adjoint if 𝒮=𝒮*. From now on, we write relation to mean linear relation.

A relation is called isometry if (4)f1,f2=g1,g2,  (f1,g1),(f2,g2) and is unitary if it is isometry and D()=R()=.

Let (z-)={(f,zf-g):(f,g)},N(,z)={f:(f,zf)} and z=R(z-).

It is clear to see that N(*,z¯)=z.

In Section 2, we establish the condition for a symmetric relation to have self-adjoint extensions in terms of defect indices and in Section 3 we discuss spectral theory.

2. Defect Indices and Self-Adjoint Extension

Let be a relation on a Hilbert space . The set (5)Γ()={z:there    exists    a    C(z)>0  such    that(zf-g)C(z)f,  (f,g)} is defined as the regularity domain of and S()=-Γ() is defined as the Spectral Kernel of . Note that

zΓ() if and only if (z-)-1 is a bounded linear operator on ;

if is symmetric, then -Γ();

Γ() is open.

The subspace z is called the defect space of and z. The cardinal number β(,z)=dimz is called the defect index of and z.

Theorem 1.

The defect index β(,z) is constant on each connected subset of Γ(). If is symmetric, then the defect index is constant on the upper and lower half-planes.

Proof.

Let Qz denote the orthogonal projection onto ¯z. We first show that Qz-Qz00 as zz0, for any z0Γ(). Let z0Γ(); then there is a constant C(z0)>0 such that (6)fC(z0)z0f-g, for all (f,g). For |z-z0|<1/(2C(z0)) and all (f,g), we have (7)fC(z0)z0f-g(zf-g+|z-z0|f)C(z0)zf-g+12ffC(z0)zf-g. For hR(Qz0)=z0, (8)Qzh=sup{|h,zf-g|:zf-gz,zf-g1}=sup{|h,(z-z0)f|:zf-gz,zf-g1}.Qzhh|z-z0|C(z0). Similarly, for hR(Qz)=z, (9)Qz0hh|z-z0|C(z0). It follows that (10)Qz-Qz02|z-z0|C(z0). Let Pz denote an orthogonal projection onto ¯z; then (11)pz-pz0=Qz-Qz02|z-z0|C(z0)0aszz0. Hence, if we choose 0<ϵ<1/(2C(z0)), then pz-pz0<1, for |z-z0|<ϵ. Therefore, dimz=dimz0. It follows that (12)β(,z)=β(,z0)    for  |z-z0|<ϵ. If is symmetric, then the upper and lower half-planes are connected subsets of Γ(); therefore, the defect index is constant there.

Letting be a symmetric relation on a Hilbert space, for z+, the defect index m=β(,z) and, for w-, the defect index n=β(,w) are written as a pair (m,n) and are called the defect indices of .

The Cayley transform of a symmetric relation on is defined by the relation (13)𝒱={(g+if,g-if):(f,g)}. Then clearly D(𝒱)=R(+i) and R(𝒱)=R(-i).

Theorem 2.

If is a symmetric relation on and 𝒱 is the Cayley transform of , then,

𝒱 is isometry;

R(I-𝒱)=D() and ={(f-g,i(f+g)):(f,g)𝒱};

is multi-valued if and only if N(I-𝒱){0}.

Proof.

( 1 ) Let (u1,v1),(u2,v2)𝒱; then ui=gi+ifi and vi=gi-ifi, for (fi,gi),i=1,2; then (14)u1,u2=g1+if1,g2+if2=g1,g2+g1,if2+if1,g2+if1,if2=g1,g2+ig1,f2-if1,g2-i2f1,f2=g1-if1,g2-if2=v1,v2.

( 2 ) It follows from the definition of Cayley transform.

( 3 ) Suppose that is multivalued; then there is g,g0 such that (0,g). It follows by definition of 𝒱 that (g,g)𝒱. Hence, gN(I-𝒱). On the other hand, let gN(I-𝒱),g0; then (g,g)𝒱; then, by (2), (0,2ig). Hence, is multivalued.

Theorem 3.

A relation 𝒱 on is the Cayley transform of a symmetric relation if and only if 𝒱 has the following properties.

𝒱 is an isometric relation.

R(I-𝒱)=D().

The relation is given by ={(f-g,i(f+g)):(f,g)𝒱}.

Proof.

If 𝒱 is the Cayley transform of , then, by Theorem 2, 𝒱 satisfies the properties (1) and (2). Conversely, supposing that 𝒱 has properties (1) and (2), we show that ={(f-g,i(f+g)):    (f,g)𝒱} is a symmetric relation.

Suppose (f1-g1,i(f1+g1)),(f2-g2,i(f2+g2)); then (15)i(f1+g1),(f2-g2)=-i(f1,f2-f1,g2+g1,f2-g1,g2). Since 𝒱 is an isometry, for any (f1,g1),(f2,g2)𝒱,f1,f2=g1,g2, this implies that (16)i(f1+g1),(f2-g2)=-ig1,g2+if1,g2-ig1,f2+if1,f2=-i(g1-f1,g2+g1-f1,f2)=f1-g1,i(f2+g2). Hence, is symmetric.

Theorem 4.

A symmetric relation is self-adjoint if and only if 𝒱 is unitary.

Proof.

We show that the is self-adjoint if and only if (17)R(+i)=R(-i)=. Since is symmetric, we always have *. Let (f,g)*; then if-g and R(-i)= imply that there is (h,k) such that k-ih=if-g. So i(f+h)=k+g, so that (f+h,i(f+h))*. That is (18)(f+h)N(*,i)=R((+i))={0}. This implies f=-hD(). Hence, is self-adjoint.

Conversely, suppose that is self-adjoint. Let (19)hR(-i)=N(*,-i)=N(,-i). So (h,-ih). But (20)-ih,h=h,ihih,h=-ih,h. Hence, we must have h=0. So R(-i)=. Similarly, R(+i)=.

Theorem 5.

Let be a closed symmetric relation on a Hilbert space and let 𝒱 denote its Cayley transform. One has the following.

𝒱 is the Cayley transform of a closed symmetric extension of if and only if the following holds. There exist closed subspaces F- of R(-i) and F+ of R(+i) and an isometric relation 𝒱~ on F+F- for which (21)𝒱={(f+h,g+k):(f,g)𝒱,(h,k)𝒱~},D(𝒱)=R(+i)=R(+i)F+,R(𝒱)=R(-i)=R(-i)F-. The spaces F+ and F- have the same dimension.

The relation 𝒱 in part (1) is unitary if and only if F-=R(-i) and F+=R(+i).

possess self-adjoint extension if and only if its defect indices are equal.

Proof.

( 1 ) Suppose that 𝒱 has the given form. Then 𝒱 is isometric relation, since for any (f+h,g+k)𝒱, we have (22)g+k2=g2+k2=f2+h2=f+h2. Hence, we can define a symmetric extension such that 𝒱 is its Cayley transform. Conversely, if 𝒱 is the Cayley transform of a symmetric extension of , then put F-=R(-i)R(-i),F+=R(+i)R(+i) and 𝒱~=𝒱|F+F-. Then we have the desired properties.

( 2 ) Here we have that 𝒱 is unitary if and only if (23)D(𝒱)=R(𝒱)=. That is, if and only if F-=R(-i) and F+=R(+i).

( 3 ) By (1) and (2), 𝒱 possess unitary extension if and only if there exists an isometry relation 𝒱~ onto R(+i)R(-i). This happens if and only if (24)dim(R(+i))=dim(R(-i)).

By definition of Cayley transform, it is clear that if 𝒱1 and 𝒱2 are the Cayley transforms of any two symmetric relations 1 and 2, then (25)12  ifandonlyif𝒱1𝒱2.

Theorem 6.

Let be a closed symmetric relation on a Hilbert space with defect indices (m,m). One has the following.

is a symmetric extension of if and only if the following holds. There are closed subspaces F+ of R(i+) and F- of R(i-) and an isometric mapping 𝒱^ of F+ onto F- such that (26)D()=D()+{g+𝒱^g:gF+}.

is self-adjoint if and only if is an m-dimensional extension of .

Proof.

( 1 ) Let 𝒱 and 𝒱 be the Cayley transforms of the closed symmetric relation and its symmetric extension , respectively. By Theorem 5, there exist closed subspaces F- of R(-i) and F+ of R(+i) and an isometric relation 𝒱~ on F+F- for which (27)𝒱={(f+h,g+k):(f,g)𝒱,(h,k)𝒱~},D(𝒱)=R(+i)=R(+i)F+,R(𝒱)=R(-i)=R(-i)F-. Then by definition of the Cayley transform, we see that (28)D()=R(I-𝒱)=(I-𝒱)D(𝒱)=(I-𝒱)R(i+)=(I-𝒱)(R(i+)F+)=(I-𝒱)(D(𝒱)F+)=(I-𝒱)D(𝒱)+(I-𝒱)F+=D()+{g-𝒱~g:gF+}. The converse is similar.

( 2 ) By Theorem 5, is self-adjoint if and only if 𝒱 is unitary. This happens if and only if F+=R(+i). So, by (1), is self-adjoint if and only if it is an m-dimensional extension of .

Theorem 7.

Suppose that 𝒯 is a self-adjoint relation and suppose that zΓ(𝒯); then (29)={zf-g:(f,g)𝒯}.

Proof.

We will show that R(z-𝒯)={zf-g:(f,g)𝒯} is a closed subspace of . Since zΓ(𝒯), there is a constant C(z) such that (30)zf-gC(z)f. Let vnR(z-𝒯) and vnvin. Suppose that fnD(𝒯) such that (fn,gn)𝒯 and vn=zfn-gn so that (fn,vn)z-𝒯. But from the above relation we have (31)vn-vm=z(fn-fm)-(gn-gm)Cfn-fm. It follows that fn is a Cauchy sequence in , and it converges to some f in . Hence, (fn,vn)(f,v). Since 𝒯 is closed, fD(𝒯) and (f,v)z-𝒯 and vR(z-𝒯). Hence, R(z-𝒯) is closed. So we have (32)=R(z-𝒯)R(z-𝒯). We next show that R(z-𝒯)={0}. Let hR(z-𝒯)=N(𝒯,z¯); then (h,z¯h)𝒯. But 0=z¯h-z¯hC(z¯)h implies h=0 a. e..

Let 𝒯 be a self-adjoint relation on and zΓ(𝒯). Define T: by T(zf-g)=f. That is T=(z-𝒯)-1={(zf-g,f):  (f,g)𝒯}. Then T is a bounded linear operator since (33)T=supzf-g=1T(zf-g)=supzf-g=1f1C(z) and 𝒯 is given by (34)𝒯={(Tf,zTf-f):f}.

3. Spectral Theory of a Linear Relation Definition 8.

Let be a closed relation on a Hilbert space . We define (35)ρ()={={(Tf,zTf-f):  f}}z:TB(),={(Tf,zTf-f):f}} to be the resolvent set and σ()=-ρ() to be the spectrum of .

Remark 9.

When a relation is an operator on , then (36)ρ()={z:(z-)-1B()}.

A complex number z is called an eigenvalue of a relation if there exists f,f0 such that (f,zf). The set of all eigenvalues of is called the point spectrum of and is denoted by σp().

Remark 10.

For any closed relation on a Hilbert space , σp()σ().

Let 𝒵={(0,g)} and Z={g:(0,g)} be the multivalued part of a relation . Clearly Z is a closed subspace of . Note that D() is not dense if is multivalued. Now define the quotient space s=/Z. We know that this quotient space is also a Hilbert space with the norm defined by (37)[f]=infgZf+g. Define a relation s on ss by s={([f],[g]):(f,g)}. We consider the relation s as the restriction of on s. By natural isomorphism, the space s is identified as Z and the relation s as 𝒵. Then clearly s is an operator on s with D(s)=D().

Theorem 11.

If 𝒯 is a self-adjoint relation on , then (38)S(𝒯)=σ(𝒯)=σ(𝒯s).

Proof.

Letting zΓ(𝒯), then there exists a constant C>0 such that (39)zf-gCf,(f,g)𝒯. For such z, we can define T=(z-𝒯)-1 as a bounded linear operator on such that 𝒯={(Th,zTh-h):h}. So zρ(𝒯). On the other hand, let zρ(𝒯); then there exists TB() such that 𝒯={(Th,zTh-h):h}. For any (f,g)𝒯, there is h such that Th=f and zTh-h=g. So (40)zf-g=zTh-zTh+h=hCTh=Cf for some C>0 and hence zΓ(𝒯). Hence, S(𝒯)=σ(𝒯).

Next, assuming that zΓ(𝒯s), then for any ([f],[g])𝒯s, there exists a constant C>0 such that (41)z[f]-[g]C[f]. For any (f,g)𝒯 we have (42)zf-gz[f]-[g]C[f]=Cf. Hence zΓ(𝒯). On the other hand, suppose that zΓ(𝒯), then there is a constant C>0 such that (43)zf-gCf. For any ([f],[g])𝒯s, we have (44)z[f]-[g]=infuZzf-g+u=infuZ(zf-g+u)zf-gCf=C[f]. This implies that zΓ(𝒯s). Thus, S(𝒯s)=S(𝒯). Hence, S(𝒯)=σ(𝒯)=σ(𝒯s).

Remark 12.

If 𝒯 is a self-adjoint relation on , then σ(𝒯).

Theorem 13.

Let zΓ(𝒯) and T=(z-𝒯)-1. One has the following.

If λΓ(T), then (z-(1/λ))Γ(𝒯).

If λS(𝒯), then (1/(z-λ))S(T).

S(T)σ(T).

Proof.

( 1 ) Let λΓ(T); then by definition there exists C(λ)>0 such that (45)λ(zf-g)-fC(λ)zf-g,(f,g)𝒯. Note that λ0. For any (f,g)𝒯, we have (46)(z-1λ)f-g=1|λ|zλf-f-λg=1|λ|λ(zf-g)-fC(λ)|λ|zf-gC(λ)C(z)|λ|f. So (z-(1/λ))Γ(𝒯).

( 2 ) Let λS(𝒯) and suppose that (1/(z-λ))S(T). Then (1/(z-λ))Γ(T). But by (1),(z-1/(1/(z-λ)))Γ(𝒯). This implies that λΓ(𝒯) which is a contradiction.

( 3 ) Let λρ(T); then (λ-T)-1 is bounded and is defined on all of ; then for any (f,g)𝒯(47)zf-g=(λ-T)-1(λ-T)(zf-g)(λ-T)-1λ(zf-g)-T(zf-g)λ(zf-g)-T(zf-g)1(λ-T)-1zf-g.λΓ(T). This shows that S(T)σ(T).

Note. We may think of developing further the spectral theory of linear relation analogous to that of operator theory. Any extension would be useful in many of the application problems which induce linear relation instead of linear operator.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Arens R. Operational calculus of linear relations Pacific Journal of Mathematics 1961 11 9 23 MR0123188 10.2140/pjm.1961.11.9 ZBL0102.10201 Coddington E. A. Extension Theory of Formally Normal and Symmetric Subspaces 1973 Providence, RI, USA American Mathematical Society iv+80 Memoirs of the American Mathematical Society, no. 134 MR0477855 Dijksma A. de Snoo H. S. V. Self-adjoint extensions of symmetric subspaces Pacific Journal of Mathematics 1974 54 71 100 MR0361889 10.2140/pjm.1974.54.71 ZBL0304.47006 Langer H. Textorius B. On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space Pacific Journal of Mathematics 1977 72 1 135 165 MR0463964 10.2140/pjm.1977.72.135 Favini A. Yagi A. Degenerate Differential Equations in Banach Spaces 1999 215 New York, NY, USA Marcel Dekker xii+313 Monographs and Textbooks in Pure and Applied Mathematics MR1654663 Hassi S. de Snoo H. Winkler H. Boundary-value problems for two-dimensional canonical systems Integral Equations and Operator Theory 2000 36 4 445 479 10.1007/BF01232740 MR1759823 ZBL0966.47012 Hassi S. Remling C. de Snoo H. Subordinate solutions and spectral measures of canonical systems Integral Equations and Operator Theory 2000 37 1 48 63 10.1007/BF01673622 MR1761504 ZBL0967.34073 Weidmann J. Linear Operators in Hilbert Spaces 1980 68 New York, NY, USA Springer xiii+402 Graduate Texts in Mathematics MR566954 Remling C. Schrödinger operators and de Branges spaces Journal of Functional Analysis 2002 196 2 323 394 10.1016/S0022-1236(02)00007-1 MR1943095 ZBL1054.34019