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Indagationes Mathematicae 23 (2012) 1087–1117
www.elsevier.com/locate/indag
Infinite-dimensional perturbations, maximally nondensely defined symmetric operators, and some
matrix representations
S. Hassi
a, H.S.V. de Snoo
b,∗, F.H. Szafraniec
caDepartment of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland bJohann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407,
9700 AK Groningen, Netherlands
cInstytut Matematyki, Uniwersytet Jagiello´nski, ul. Łojasiewicza 6, 30 348 Krak´ow, Poland
To the memory of Israel Gohberg
Abstract
The notion of a maximally nondensely defined symmetric operator or relation is introduced and characterized. The selfadjoint extensions (including the generalized Friedrichs extension) of a class of maximally nondensely defined symmetric operators are described. The description is given by means of the theory of ordinary boundary triplets and exhibits the extensions as infinite-dimensional perturbations of a certain selfadjoint operator extension of the symmetric operator.
c
⃝2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
Keywords:Symmetric operator; Generalized Friedrichs extension; Weyl function; Boundary triplet; Graph perturbation
1. Introduction
As an illustration of the topics in this paper consider the following situation. Let S be a bounded, closed, symmetric operator in a Hilbert space H. Then H has the orthogonal
∗Corresponding author. Tel.: +31 503633963.
E-mail addresses:sha@uwasa.fi(S. Hassi),desnoo@math.rug.nl(H.S.V. de Snoo),umszafra@cyf-kr.edu.pl (F.H. Szafraniec).
0019-3577/$ - see front matter c⃝2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.indag.2012.08.007
decompositionH=H1⊕H2withH1=domSand dimH2>0, and
S f1=S11f1⊕S21f1, Sj1∈B(H1,Hj), j =1,2, f1∈H1. (1.1) Referring to this formula let the matrix
A=df
S11 S∗21 S21 S22
, (1.2)
corresponding to the decompositionH=H1⊕H2, be an arbitrary bounded selfadjoint operator extension of S. Of course S will also have unbounded selfadjoint operator extensions and, in fact,S will have selfadjoint extensions, which are relations, i.e., multivalued linear mappings.
The following result may serve as a starting point to approach the main topics of the paper;
cf. [21, Proposition 5.1] and [12, Proposition 3.5] for an inverted form. Here and elsewhere the language of boundary triplets will be used freely; cf. [11,12].
Proposition 1.1.Let S be a closed bounded symmetric operator as in(1.1)and(1.2). Then the following statements hold:
(i) S has equal defect numbers(d,d),d =dimH2≤ ∞; (ii) the adjoint S∗of S inHis the relation given by
S∗= {f =(f,A f +h); f ∈H,h∈H2};
(iii) a boundary triplet for S∗is given byΠ = {H2,Γ0,Γ1}, where
Γ0f = −h, Γ1f = f2; f = f1⊕ f2∈H, f =(f,A f +h)∈ S∗;
(iv) the correspondingγ-fieldγ is given byγ (λ) =(A−λ)−1 H2and the Weyl function M is given by
M(λ)=P2(A−λ)−1 H2=
S22−λ−S21(S11−λ)−1S21∗
−1
, where P2stands for the orthogonal projection ontoH2;
(v) the selfadjoint extensions AΘ =ker(Γ0+ΘΓ1)of S inHare in one-to-one correspondence with the selfadjoint relationsΘinH2via
AΘ =
S11 S21∗ S21 S22+Θ
,
and their resolvents, withλ∈ρ(AΘ)∩ρ(A), are connected by (AΘ−λ)−1=(A−λ)−1−γ (λ)(M(λ)+Θ−1)−1γ (λ)¯ ∗.
Note thatS∗is a relation with the multivalued part mulS∗ =H2. The mappingsΓ0andΓ1 generate selfadjoint extensions ofSas follows:
kerΓ0=A, kerΓ1=S11⊕({0} ×mulS∗),
and they are transversal:S∗=kerΓ0+kerΓ1, where the sum is componentwise. The selfadjoint extension kerΓ1corresponds to the choiceΘ = {0} ×H2. IfSis semibounded, then kerΓ1is the usual Friedrichs extension ofS. Since ranγ (λ)=ker(S∗−λ), an application of the Frobenius formula for the inverse(A−λ)−1yields
P2ker(S∗−λ)= {(S22−S21(S11−λI1)−1S21∗ −λI2)−1h; h∈H2},
where P2 stands for the orthogonal projection ontoH2. This formula implies that the defect subspaces of the bounded nondensely defined symmetric operatorSadmit the following property
ker(S∗−λ)∩domS = {0} for allλ∈C\R. (1.3)
The present paper deals with the general class of symmetric, not necessarily closed, relationsS in a Hilbert spaceHsatisfying(1.3). Such relations will be calledmaximally nondensely defined;
cf.Definition 3.3. The property(1.3)holds precisely when the componentwise sum
S∞=S+({0} ×mulS∗) (1.4)
is an essentially selfadjoint extension of S; see Proposition 3.10. Thus the closure of S∞ is a selfadjoint extension of S inH. In particular, this means that S has equal defect numbers and clearly, if S is not itself essentially selfadjoint, then by taking closures in (1.4) one concludes that mulS∗ ̸= mulS∗∗. Consequently, the defect numbers (d,d) are nonzero as d =dim(mulS∗⊖mulS∗∗); seeLemma 3.2. The extensionS∞is selfadjoint if and only if
domS=domS∩domS∗; (1.5)
a condition, which has appeared earlier in [8]. Various characterizations for symmetric relations satisfying(1.4)or(1.5)will be established; cf.Propositions 3.10and3.11.
One of the aims of the paper is to present an extension ofProposition 1.1which will now be sketched; cf.Theorem 5.1andCorollary 5.2. Let S be a closed symmetric operator, which satisfies(1.5)so that it is maximally nondensely defined, and letAbe some selfadjoint operator extension ofS, which is transversal toS∞(there always exist such extensions AofS). Then the selfadjoint extensionsAΘofSare in one-to-one correspondence with the selfadjoint relationsΘ inHvia the perturbation formula
AΘ =A+GΘG∗P2, (1.6)
whereGis a bounded and boundedly invertible operator from the Hilbert spaceHonto mulS∗; seeTheorem 5.3. The formula(1.6)admits essentially the same simplicity as the block formula in Proposition 1.1. In general, when S is not closed and bounded, one cannot rewrite the perturbation formula(1.6)as a block form; seeExample 5.6. However, such a block formula is still possible for unbounded symmetric operators, which are partially bounded; see Sections4 and5.3. Notice also that in (1.6)the operator extensions of S are parameterized by operators Θ inH, i.e., by bounded or unbounded (range) perturbations of the selfadjoint operator A.
On the other hand, the purely multivalued selfadjoint relation{0} ×HinHcorresponds to the selfadjoint extensionS∞in(1.4). In fact,S∞is the only selfadjoint extension ofSwhose domain is contained in dom Sand, hence, it coincides with the so-called generalized Friedrich extension of S; cf. [20]. If S is semibounded thenS∞ coincides with the standard Friedrichs extension;
cf. [8]. Perturbation formulas as in(1.6)often occur in concrete applications; see [1,3,9,10,15, 13,25].
One may view the underlying symmetric operator S as a domain restriction of any of its selfadjoint operator extensions AΘ in(1.6). A completely formal reasoning leads to analogous results for range restrictions S of a selfadjoint operator A. All the selfadjoint extensions of S can again be described by means of an ordinary boundary triplet which is constructed in Proposition 5.11, which gives rise to an explicit domain perturbation formula for all selfadjoint extensions ofSanalogous to(1.6). Such results find applications in problems involving ordinary and partial differential equations; see [1,14,17,28]. A typical case of this situation occurs for
symmetric densely defined operators, for whichλ=0 is a point of regular type (ranSis closed, S−1is bounded), or if for instance there is a selfadjoint extension ofS with discrete spectrum.
Roughly speaking all what is needed to derive such results is to pass to the formal inverseS−1, which in turn becomes (the graph of) a maximally nondensely defined symmetric relation, and then describe the inversesA−1Θ of the selfadjoint extensionsAΘofSas the range perturbation of S−1as in(1.6).
The contents of the paper are as follows. In Section2some preliminary results concerning range perturbations and ordinary boundary triplets are presented; see [16] for linear relations and [12,17] for ordinary boundary triplets. The class of maximally nondensely defined symmetric relations is investigated in Section3 and continued further in Section 4 under the additional assumption thatSis partially bounded. The extension theory of maximally nondensely defined symmetric relations is presented in Section5with a construction of suitable boundary triplets for S∗ yielding in particular the (range) perturbation formula as in (1.6). In that section also connections to block matrix formulas are given covering the known special case of bounded symmetric operators. Finally, the translation to the perturbations on the side of domains is shortly described.
2. Preliminaries
2.1. Some facts about linear relations
Let S be a, not necessarily closed, linear relation in a Hilbert spaceH, with inner product
⟨,·,·⟩, so thatH=dom S∗⊕mulS∗∗. The so-called (orthogonal)operator partof Ais defined by
Sop= {df (f, f′)∈S; f′∈dom S∗}. (2.1)
Let Q stand for the orthogonal projection from H onto domS∗. Then S admits a so-called canonical decomposition into the (operator wise) sum of its regular and singular parts S = Sreg+Ssingwhich are defined as follows:
Sreg
=df Q S= {(f,Q f′);(f, f′)∈S}, (2.2)
Ssing=df (I−Q)S= {(f, (I−Q)f′);(f, f′)∈S}; (2.3) see [23]. By definition domSreg = domSsing = domS and, moreover, the regular partSreg is a regular, i.e. closable, operator and the singular partSsingis a singular relation, i.e. ranSsing ⊂ mulSsing∗∗ .
Then(2.1)and(2.2)show that Sop ⊂ Sreg, which implies that as a restriction of a closable operator, the operator part of S is also closable. For further properties of this and some other related decompositions of linear relations the reader is referred to the papers [23,16].
Let S be a, not necessarily closed, symmetric linear relation in a Hilbert spaceH, so that S ⊂S∗, or equivalently,⟨f′, f⟩ ∈R. The closureS∗∗is also symmetric with the same adjoint S∗, thusS⊂S∗∗⊂S∗. For a symmetric relationSone has
ran(S−λ)∩mulS∗=mulS, λ∈C\R. (2.4)
To see this, it suffices to show that the left-hand side is contained in the right-hand side. Let h ∈ran(S−λ)∩mulS∗, thenh = f′−λf for some(f,f′)∈ S; hence⟨f′, f⟩ =λ⟨f, f⟩, which leads to f =0 andh= f′∈mulS.
Thedefect subspacesofSare defined byNλ(S∗)=(ran(S− ¯λ))⊥=ker(S∗−λ). Thedefect numbersofSare defined by dim ker(S∗−λ) (≤∞)and are constant forλ∈C+and forλ∈C−. The following notation is used:
Nλ(S∗)= {df fλ=df(fλ, λfλ); fλ∈Nλ(S∗)}, λ∈C. (2.5) The adjointS∗has the following componentwise sum decomposition
S∗=S∗∗+Nλ(S∗)+Nλ¯(S∗), λ∈C\R (2.6) due to von Neumann. Observe thatH=domS∗⊕mulS∗∗and decompose the closed symmetric relationS∗∗accordingly:
S∗∗=(S∗∗)op⊕({0} ⊕mulS∗∗), (2.7) where the orthogonal operator part(S∗∗)op =Q S∗∗ =Sreg∗∗ is as in(2.1),(2.2); for more details on such decompositions see also [16]. Taking adjoints one obtains
S∗=((S∗∗)op)∗⊕({0} ⊕mulS∗∗), which leads in particular to
ker(S∗−λ)=ker(((S∗∗)op)∗−λ), λ∈C. (2.8)
mulS∗⊖mulS∗∗=mul((S∗∗)op)∗. (2.9)
Thus the deficiency indices of S inHare equal to the ones of its operator part(S∗∗)op in the subspaceH⊖mulS∗∗.
2.2. Range perturbations of linear relations
Recall that for closed subspacesMandNof a Hilbert spaceHthe sumM+Nis closed if and only ifM⊥+N⊥is closed; see [24, IV, Theorem 4.8]. The following lemma is a weakening of a known result; cf. [12,16].
Lemma 2.1. Let M be a linear, not necessarily closed, subspace, let N be a closed linear subspace of a Hilbert spaceH, and let P be the orthogonal projection fromHontoN⊥. Then
(i) M+Nis closed if and only if PMis closed;
(ii) M+N=Hif and only if PM=N⊥; (iii) ker(P M)=M∩N.
Proof. SinceNis closed, the Hilbert spaceHallows the orthogonal decompositionH=N⊕N⊥. Now it is straightforward to check the following identity:
M+N= PM⊕N.
This implies immediately the statements in (i)–(ii). Statement (iii) is clear.
In the sequel also the following closely related fact is needed.
Lemma 2.2. LetMandNbe linear, not necessarily closed, subspaces of a Hilbert spaceH, and let P be the orthogonal projection fromHonto the closed subspaceN⊥. ThenM+Nis dense inHif and only if PMis dense inN⊥.
Proof. Observe the following inclusions:
M+N⊂M+closN=PM⊕closN⊂clos(M+N).
Hence clos(PM)⊕closN=clos(M+N), which implies the claim.
For the calculus of linear relations in a Hilbert space, involving adjoints and componentwise sums, see for instance [11,16].
Lemma 2.3. Let S and T be closed linear relations from a Hilbert spaceHto a Hilbert space K. Then the following statements are equivalent:
(i) S+T is a closed relation fromHtoK;
(ii) S∗+T∗is a closed relation fromKtoH.
Proposition 2.4.Let T be a relation from a Hilbert spaceHto a Hilbert spaceK, letLbe a closed linear subspace inK, and let PLbe the orthogonal projection fromKontoL. Then the following statements are equivalent:
(i) T∗∗+({0} ×L)is closed inH×K;
(ii) domT∗+L⊥is closed inK;
(iii) domT∗+L⊥=(mulT∗∗∩L)⊥; (iv) PL(domT∗)is closed inK.
Moreover, if mulT∗∗+Lis closed inKor, equivalently, if domT∗+L⊥is closed inK, then each of the statements(i)–(iv)is equivalent to each of the following statements:
(v) domT∗⊂domT∗+L⊥; (vi) PL(domT∗)=PL(domT∗).
In particular, if T is a closable operator from H to K, i.e., if mulT∗∗ = {0}, then the statements(v)and(vi)reduce to:
(vii) domT∗+L⊥=K;
(viii) PL(domT∗)=L, respectively.
Proof. (i)⇔(ii) The assumption thatT∗∗+({0}×L)is a closed subspace inH×Kis equivalent, viaLemma 2.3, to the closedness of the subspace
T∗+(L⊥×H)=(domT∗+L⊥)×H,
which is closed precisely when domT∗+L⊥is closed.
(ii)⇔(iii) Observe that mulT∗∗=(domT∗)⊥, which implies that (domT∗+L⊥)⊥=mulT∗∗∩L,
sinceLis closed. Therefore,
clos(domT∗+L⊥)=(mulT∗∗∩L)⊥, (2.10)
which shows that domT∗+L⊥is closed if and only if (iii) is satisfied.
(ii)⇔(iv) ApplyLemma 2.1withM=domT∗,N=L⊥, andP=PL.
Next observe that the subspace mulT∗∗+Lis closed if and only if the subspace(mulT∗∗)⊥+ L⊥=domT∗+L⊥is closed, and that in this case
(mulT∗∗∩L)⊥=domT∗+L⊥. (2.11)
(iii)⇔(v) When mulT∗∗+Lis closed, it follows from(2.11)that the condition domT∗+ L⊥=(mulT∗∗∩L)⊥can be rewritten as
domT∗+L⊥=domT∗+L⊥, which is equivalent to
domT∗⊂domT∗+L⊥.
(v) ⇒ (vi) It is clear that in general PL(domT∗) ⊂ PL(domT∗). The reverse inclusion follows directly from (v) as dom T∗⊂domT∗+L⊥. Hence (vi) holds.
(vi)⇒(v) Let f ∈ domT∗. Then there existsh ∈ domT∗ such that f −h ∈ L⊥ and f =h+(f −h)∈domT∗+L⊥. Hence (v) holds.
Finally observe that ifT is a closable operator, then(domT∗)⊥=mulT∗∗= {0}. Hence, in this case mulT∗∗+L=Lis closed and furthermore domT∗=K. Therefore the statements (v) and (vi) clearly reduce to the statements (vii) and (viii), respectively.
Remark 2.5. In [12, Definition 2.1] a linear relationT satisfying the property stated in (i) of Proposition 2.4has been calledL-regular; in [12, Proposition 2.5] the equivalence of (i) and (iv) inProposition 2.4has been also proved.
2.3. Ordinary boundary triplets
LetS be a closed symmetric relation in a Hilbert spaceH. If the defect numbers are equal, thenShas selfadjoint extensions in the Hilbert spaceH. LetA0andA1be selfadjoint extensions ofS; they are calleddisjointwith respect toS ifA0∩A1=Sandtransversalwith respect toS ifA0+A1=S∗; see [12, Definition 1.7]. Some further definitions and facts which can be found in [12] are now given.
Definition 2.6 ([12]).LetS be a symmetric relation in a Hilbert spaceHwith equal deficiency indices and letS∗be its adjoint. Then the tripletΠ =(H,Γ0,Γ1), whereHis a Hilbert space andΓ =df (Γ0,Γ1)is a linear single-valued surjection ofS∗ontoH2=H×H, is said to be an ordinary boundary tripletforS∗if the abstract Green’s identity
⟨f′,g⟩H− ⟨f,g′⟩H= ⟨Γ1f,Γ0g⟩H− ⟨Γ0f,Γ1g⟩H, (2.12) holds for all f =(f, f′),g =(g,g′)∈ S∗. IfS is closed one may think of(H,Γ0,Γ1)as the ordinary boundary triplet ofSitself.
If(H,Γ0,Γ1)is a boundary triplet for S∗, then dimH = n±(S). Moreover, S = kerΓ ⊂ kerΓ0∩kerΓ1and the relationsA0andA1defined by
A0df
=kerΓ0, A1df
=kerΓ1, (2.13)
are selfadjoint extensions of S and they are transversal with respect to S. Conversely, for any two selfadjoint extensionsA0andA1ofSwhich are transversal with respect toS, there exists a boundary triplet(H,Γ0,Γ1)forS∗such that(2.13)holds. In particular, a boundary triplet is not unique if the defect numbers ofSare not equal to zero.
Boundary triplets are particularly convenient for the parameterization and description of the intermediateextensions H ofS, i.e., the extensions Hof S which satisfyS ⊂ H ⊂ S∗. More
precisely, the mapping
Θ → AΘ = {df f ∈S∗;Γf ∈Θ} =ker(Γ1−ΘΓ0) (2.14) establishes a bijective correspondence between the closed relations Θ in H and the closed intermediate extensionsAΘofS. Furthermore, it can be shown that
AΘ∗ =(AΘ)∗. (2.15)
In particular, a closed extensionAΘ of Sis symmetric or selfadjoint if and only if the relation Θis symmetric or selfadjoint, respectively. A specific symmetric subspace inH, which is called in [12, p. 141] aforbidden manifold, is defined as follows:
FΠ =dfΓ({0} ×mulS∗). (2.16)
Note that the symmetric extension ofScorresponding toFΠ in(2.14)is given by S∞
=df AFΠ =S+({0} ×mulS∗);
this extension has an important role in later sections. Finally recall that the intermediate extensions in(2.14)have the following properties (see [12, Proposition 1.4]):
AΘ∩A0=S(disjoint)⇐⇒Θoperator, (2.17)
and
AΘ+A0=S∗(transversal)⇐⇒Θ bounded operator. (2.18) Definition 2.7([12]).Let(H,Γ0,Γ1)be a boundary triplet forS∗withA0=kerΓ0. TheΓ-field γis defined by (see(2.5))
γ (λ)= {(Γ0fλ,fλ); fλ∈Nλ(S∗)}, λ∈ρ(A0), (2.19) and theWeyl function Mis defined by
M(λ)= {(Γ0fλ,Γ1fλ); fλ∈Nλ(S∗)}, λ∈ρ(A0). (2.20) Denote byΠ1the orthogonal projection inH⊕Honto the first component. Observe that the restrictionΓ0Nλ(S∗)of the mappingΓ0toNλ(S∗)is a bijective mapping ontoH. Hence,γ (λ) is the graph of a bounded linear operator fromHtoNλ(S∗)andM is aB(H)-valued function, given by
γ (λ)=Π1(Γ0Nλ(S∗))−1, M(λ)=Γ1(Γ0Nλ(S∗))−1, λ∈ρ(A0).
For allλ, µ∈ρ(A0)theΓ-fieldγ satisfies the identity
γ (λ)=(I +(λ−µ)(A0−λ)−1)γ (µ), (2.21)
which, in particular, shows thatγis holomorphic onρ(A0). The Weyl functionMand theΓ-field γare related via the identity
M(λ)−M(µ)∗
λ− ¯µ =γ (µ)∗γ (λ), λ, µ∈ρ(A0). (2.22)
Since γ (λ) is injective and maps H onto Nλ(S∗), (2.22) shows that ImM(λ) is boundedly invertible. By means ofM the identity(2.14)can be rewritten as
(AΘ−λ)−1=(A0−λ)−1−γ (λ)
M(λ)−Θ−1
γ (λ)¯ ∗. (2.23)
The classN0(H)is introduced as the collection of all Nevanlinna functionsM, which satisfy sup
y>0
y⟨ImM(iy)h,h⟩<∞, (2.24)
for allh∈H; cf. [20]. In this case there exists a selfadjoint operatorE∈B(H)such that
λlim→∞M(λ)h=E h, h ∈H.
It is well-known that for any Nevanlinna functionMthere exists an operatorB∈B(H)such that
λlim→∞
M(λ)h
λ =Bh, h∈H. IfBis boundedly invertible then
s−lim
y↑∞iy M(iy)−1=B−1. (2.25)
A Nevanlinna function is said to beuniformly strict, when 0∈ρ(ImM(λ))for allλ∈C\R; in this case also−M(λ)−1belongs toN(H)and is uniformly strict. The boundedness of the limit value in(2.25)shows that the function−M(λ)−1belongs to the subclassN0(H).
3. Maximally nondensely defined symmetric relations
3.1. Orthogonal projections of nondensely defined symmetric relations
LetSbe a symmetric relation in a Hilbert spaceHand assume it is nondensely defined so that mulS∗is nontrivial. The Hilbert spaceHadmits the orthogonal decompositionH =H1⊕H2 withH1 = domS andH2 = mulS∗. Note that H1 ⊂ domS∗. Let P denote the orthogonal projection fromHonto mulS∗. Note that mulS∗∗ ⊂mulS∗so that(mulS∗)⊥ ⊂(mulS∗∗)⊥. Define the linear relationsS1=(I−P)SandS2=P SinHby
S1= {(f, (I−P)f′);(f, f′)∈ S}, S2= {(f,P f′);(f,f′)∈S}. (3.1) It is clear that domS1 = domS ⊂ H1, ranS1 ⊂ dom S = H1, and that S1 is a symmetric operator inH. Moreover,
S1∗=S∗(I−P), S1∗∗=(S∗(I−P))∗⊃(I−P)S∗∗.
Note that mulS∗ = ker(I − P) ⊂ domS1∗. Likewise it is clear that domS2 = domS ⊂ H1,ranS2⊂mulS∗=H2, and thatS2is a relation inHwith mulS2=mulS. Moreover,
S2∗=S∗P, S∗∗2 =(S∗P)∗⊃P S∗∗.
Note that domS =kerP ⊂domS∗2, which implies that
domS2∗=domS∗=H⊖mulS∗∗=domS⊕(mulS∗⊖mulS∗∗), (3.2) in particular, mulS2∗∗=mulS∗∗.
In some senseS1andS2resemble the regular and singular parts of the relationS. However it will be useful to consider them in spaces related to the decompositionH=H1⊕H2. The relation S1is a linear subspace of the productH×H. However, the inclusions domS1⊂H1,ranS1⊂H1 show thatS1may also be considered as a linear subspace of the productH1×H1; therefore define
S11=S1∩(H1×H1), (3.3)
in other wordsS11 as a graph is the same asS1 but considered in the product spaceH1×H1, rather than in the product space H× H. Clearly S11 is a symmetric operator in H1 with domS11=domS; henceS11is densely defined and closable inH1. Observe that
(S11)∗=S∗∩(H1×H1), (3.4)
so that S11∗ ⊂ S∗. The relation S2 is a linear subspace of the productH×H. However, the inclusions domS2⊂H1,ranS2⊂H2show thatS2may also be considered as a linear subspace of the productH1×H2; therefore define
S21=S2∩(H1×H2), (3.5)
in other wordsS21 as a graph is the same asS2 but considered in the product spaceH1×H2, rather than in the product spaceH×H. Clearly domS21 = domS, so that the relation S11 is densely defined inH1. Observe that
(S21)∗=S∗∩(H2×H1), (3.6)
so thatS21∗ ⊂S∗. The adjoints in(3.4)and(3.6)ofS11andS21have been taken with respect to the spacesH1×H1andH1×H2, respectively. This convention will be used in the rest of the paper;
in fact, the notationsSi j∗ andSi j∗∗,i,j =1,2, will stand for(Si j)∗and(Si j)∗∗, respectively. The next lemma can be derived from [22]; see also [30,27]. Observe that if the multi-valued relation Sis not symmetric, then the following matrix representation fails to hold in general.
Lemma 3.1. Let S be a symmetric relation in the Hilbert spaceH. Then S admits the following block representation as a multi-valued column operator fromH1toH1⊕H2with entries S11and S21defined in(3.3),(3.5):
S =
S11 S21
. (3.7)
Moreover, the adjoint S∗is a, not necessarily densely defined, operator fromH1⊕H2toH1such that
S∗⊃
S11∗ S21∗
, S∗∗⊂
S∗11 S21∗∗
=
S∗∗11 S∗∗21
. (3.8)
Conversely, if S11 is a symmetric operator inH1and S21is a linear relation fromH1toH2 such thatdomS11∩domS21is dense inH1, then S defined by(3.7)is a symmetric relation inH with a dense domain inH1.
Proof. Clearly the inclusion “⊂” in (3.7)holds for an arbitrary (not necessarily symmetric) relationS. Since S is symmetric, mulS ⊂ mulS∗ = H⊖domS = H2 and hence mulS = mulS21; this implies that the inclusion “⊃” in(3.7)is also satisfied. The formulas in(3.8)follow from(3.7)using the general result on block relations in [22, Proposition 2.1].
The converse statement can be checked in a straightforward manner.
In particular, Lemma 3.1 implies that S is closable if and only if S21 is closable and that mulS21∗∗=mulS∗∗or, equivalently,
domS21∗ =mulS∗⊖mulS∗∗=domS∗⊖H1. (3.9)
If the adjoint of S is calculated in Hthen it is given by S∗+({0} ×mulS∗), where S∗ is the adjoint as inLemma 3.1.
The defect subspaces of the densely defined symmetric operatorS11inH1are said to bethe semidefect subspacesof the original symmetric relationSinH: they are denoted by
Nλ(S11∗)=ker(S11∗ −λ)=H1⊖ran(S11− ¯λ), λ∈C\R. (3.10) The dimensions of the semidefect subspaces ofS, i.e. the defect numbers ofS11inH1, are called the semidefect numbers of S; see [2,31, Section 1.5]. To formulate the next lemma denote byPλ the orthogonal projection fromHonto ker(S∗−λ), so that kerPλ=ran(S− ¯λ).
Lemma 3.2. Let S be a symmetric relation in a Hilbert spaceH. Then
Nλ(S11∗)=Nλ(S∗)∩dom S, λ∈C\R, (3.11)
and moreover,
Pλ(Nλ(S11∗)⊕mulS∗)=Nλ(S∗), (3.12)
ker(Pλ (Nλ(S11∗)⊕mulS∗))=mulS∗∗. (3.13) In particular,
dimNλ(S∗)=dimNλ(S11∗ )+dim(mulS∗⊖mulS∗∗), λ∈C\R.
Proof. The identity(3.11)follows directly from(3.4). It follows from(3.10)that
ran(S11− ¯λ)⊕Nλ(S11∗)⊕mulS∗=H. (3.14)
Observe thatS1=(I−P)Sleads to
ran(S11− ¯λ)⊂ran(S− ¯λ)+mulS∗. (3.15)
Hence a combination of(3.14)and(3.15)gives ran(S− ¯λ)+(Nλ(S11∗)⊕mulS∗)=H,
which leads to(3.12). Finally apply(2.4)withS∗∗to obtain(3.13).
It follows fromLemma 3.2that for a nondensely defined symmetric relationSthere are two extreme cases. The one extreme case occurs when Nλ(S∗11) = {0}or, equivalently, when S11 is essentially selfadjoint inH1. Since this case has an important role in the present paper, the following definition is introduced.
Definition 3.3. A symmetric, not necessarily closed, relation S is said to be maximally nondensely defined if the semidefect numbers of S are equal to(0,0), or equivalently, if the following equality holds:
Nλ(S∗)∩domS= {0}, λ∈C\R.
The other extreme case occurs when Nλ(S∗11) = Nλ(S∗) or, equivalently, whenNλ(S∗) ⊂ domS. For the sake of completeness this last case will be described in the rest of the present subsection.
Lemma 3.4. Let S be a symmetric relation in a Hilbert spaceH. Then the following statements are equivalent:
(i) Nλ(S∗)⊂domS for some (equivalently for every)λ∈C\R; (ii) mulS∗∗=mulS∗;
(iii) mulS21∗∗=mulS∗or, equivalently, S21is singular.
If the relation S is a closable operator, then(ii)is equivalent to (iv) domS is dense inH.
If the relation S is closed, then each of (i), (ii), or(iii)is equivalent to (v) S=Sop+({0} ⊕mulS∗).
Proof. (i)⇔(ii) In view of(3.11)one has
Nλ(S∗)⊂domS⇔Nλ(S11∗)=Nλ(S∗), λ∈C\R.
In this case it follows from
H=ran(S11− ¯λ)⊕Nλ(S11∗)⊕mulS∗=ran(S− ¯λ)⊕Nλ(S∗), that
ran(S11− ¯λ)⊕mulS∗=ran(S− ¯λ), and hence
mulS∗⊂ran(S− ¯λ)⊂ran(S∗∗− ¯λ)=ran(S∗∗− ¯λ),
which combined with(2.4)shows that mulS∗⊂mulS∗∗, i.e., mulS∗=mulS∗∗. Conversely, observe that(3.11)and(2.4)imply that
Nλ(S∗11)=Pλ(Nλ(S11∗))=Pλ(Nλ(S11∗) ⊕ mulS∗∗).
Hence, if mulS∗=mulS∗∗, then(3.12)implies that Nλ(S∗11)=Pλ(Nλ(S11∗) ⊕mulS∗)=Nλ(S∗), which with(3.11)shows thatNλ(S∗)⊂domS.
(ii)⇔(iii) Since mulS21∗∗=mulS∗∗, this means that (ii) is equivalent to mulS21∗∗=mulS∗. IfSis a closable operator, then mulS∗∗ = {0}, and condition (ii) means that mulS∗ = {0}, which is equivalent with (iv). IfSis closed, then (ii)⇔(v) follows from(2.7).
Example 3.5.The conditionNλ(S∗)⊂domSmay be satisfied withoutSbeing densely defined.
Let S21 be a singular operator in a Hilbert space of the form H1⊕H2 with H1 = domS1
andH2 = ran S2 ̸= {0}; such an operator is easily constructed from any singular operator in a, necessarily infinite-dimensional, Hilbert spaceH; cf. [16]. Let S11 be a bounded selfadjoint operator acting in H1. Then S defined by(3.7)is a symmetric operator in H with domS = domS21and mulS∗∗=mulS21∗∗=H2=(domS)⊥. Hence byLemma 3.4Nλ(S∗)⊂dom Sfor allλ∈C\R, while the operatorSis not densely defined inH.
Remark 3.6. Example 3.5shows that the conditionNλ(S∗)⊂domS, λ∈C\R, does not imply thatSis densely defined inHwhenSis not closed.
Notice also that inLemma 3.4(i) it suffices to consider the inclusionNλ(S∗)⊂domSfor one pointλ∈C\R. If one knows in addition thatNλ(S∗)⊂domSandNλ¯(S∗)⊂dom Sfor some λ∈C\R, then from the first von Neumann’s formula one concludes that domS∗⊂domS. Since Sis closable if and only ifS∗is densely defined, it follows thatS, together withS∗, is densely defined; hence using two points one obtains a more direct proof for item (iv) inLemma 3.4via von Neumann’s formula.
3.2. Some characterizations of symmetric relations
LetSbe a symmetric relation in a Hilbert spaceHand assume it is nondensely defined so that mulS∗is nontrivial. Then the relationS∞defined by the forbidden manifold (see(2.16))
S∞=S+({0} ×mulS∗), (3.16)
is a symmetric extension ofS:
S⊂S∞⊂(S∞)∗⊂S∗. (3.17)
Observe that the operatorwise sum S1+S2is a symmetric extension of S, which is contained inS∞. It follows from mulS ⊂mulS∗that mulS∞ =mulS∗. Clearly,S ⊂closS = S∗∗and, since(domS∗∗)⊥=mulS∗, it is easy to see that the following inclusions hold:
S∞⊂(S∗∗)∞⊂clos(S∞)=clos(S∗∗)∞. (3.18)
Observe that
ran(S∞−λ)=ran(S−λ)+mulS∗, λ∈C. (3.19)
Due to(2.4)the sum in(3.19)is direct forλ∈C\Rif and only ifSis an operator. The operator S11in(3.3)and the relationS∞in(3.16)are related by
S∞=S11⊕({0} ×mulS∗), (3.20)
where{0}×mulS∗is a selfadjoint relation in the Hilbert space mulS∗. The next lemma describes S∞, its adjoint, and its closure; cf. [16].
Lemma 3.7. Let S be a symmetric relation in a Hilbert spaceH. Then
S∞= {(f,f′)∈S∗; f ∈domS}, (3.21)
(S∞)∗= {(f, f′)∈S∗; f ∈domS}, (3.22)
and
(S∞)∗∗= {(f, f′)∈S∗; f ∈domS11∗∗}. (3.23) Proof. The formulas(3.21)and(3.22)can be found in [16,8]. To prove(3.23), observe that the orthogonal sum decomposition in(3.20)implies that
(S∞)∗∗=S11∗∗⊕({0} ×mulS∗). (3.24)
In particular,S11∗∗⊂(S∞)∗∗⊂S∗. Now it is easy to check that the formula(3.24)is equivalent to the formula(3.23).
Note that(S∞)∗ = ((S∗∗)∞)∗; see(3.18). Since S is symmetric, Lemma 3.7implies that dom(S∞)∗=dom Sand, hence, mul clos(S∞)=mulS∗=mulS∞.
Lemma 3.8. Let S be a symmetric relation in a Hilbert spaceH. Then the following statements are equivalent:
(i) S∞is closed;
(ii) S11in(3.1)is a closed operator indomS;
(iii) ran(S−λ)+mulS∗is closed for some (and hence for all)λinC\R; (iv) domS=domS∗∗anddomS∗=domS∗+domS.
If the symmetric relation S is closed, then each of the statements(i)–(iv)is also equivalent to the following statement:
(v) PλmulS∗is a closed subspace of ker(S∗−λ)for some (and hence for all)λ∈C\R. Proof. (i)⇔(ii) This equivalence follows from(3.20).
(i)⇔(iii) The symmetric relationS∞is closed if and only if ran(S∞−λ)is closed for some (and hence for all)λ∈C\R. Then recall the identity in(3.19).
(i)⇒(iv) SinceS∞is closed, one hasS∞ =(S∗∗)∞; see(3.18). This shows that domS = domS∗∗. Next applyProposition 2.4withT = S∗∗andL =mulS∗. SinceS is symmetric, it follows that mulS∗∗⊂mulS∗, so that mulS∗∗+L=Lis closed. ByProposition 2.4the relation (S∗∗)∞in(3.16)is closed if and only if
domS∗⊂domS∗+(mulS∗)⊥=domS∗+domS.
Since the right-hand side is clearly contained in domS∗, this shows that (iv) holds.
(iv)⇒(i) If dom S∗ =domS∗+domS thenProposition 2.4shows that(S∗∗)∞is closed.
On the other hand, mulS ⊂mulS∗∗⊂mulS∗and hence the equality domS =domS∗∗implies the equalityS∞=(S∗∗)∞and, thus, (i) is satisfied.
Now assume in addition thatSis closed. Then equivalently ran(S−λ), λ∈C\R, is closed.
(iii)⇔(v) ApplyLemma 2.1to the closed subspacesM = mulS∗andN = ran(S− ¯λ). Note thatN⊥=ker(S∗−λ)=ranPλ. Hence, byLemma 2.1(i)
ran(S− ¯λ)+mulS∗, λ∈C\R, is closed if and only ifPλmulS∗is closed.
Remark 3.9. In the case that S is a closed operator the equivalence of (ii) and (v) in Lemma 3.8is also proved, for instance, in [2, Theorem 1.5.4] and in fact it goes back to M.A.
Krasnoselski˘ı [26]. The equivalence of (i) and (ii) has been also proved e.g. in [12, Proposition 2.5]. Some further equivalent conditions, based on the rigged Hilbert space generated by S∗ can be found in [2, Theorem 2.4.1]. In [29] (see also [2,12]) a symmetric operator S is said to beregular, if the subspaces PλmulS∗, λ ∈ C\ R, are closed; in the opposite case S is said to besingular. In the present paper the terms regular and singular refer to a more general terminology which appears in connection with general linear operators and relations in Hilbert spaces, cf. Section2; for further details see [16] and the references therein. Finally, observe that in [2, Section 2.4] the termO-operatorstands for a closed symmetric operator whose semidefect numbers are equal to (0,0), instead of the present expression “S is maximally nondensely defined” that appears inDefinition 3.3.
The next result gives several criteria forSto be maximally nondensely defined inH.
Proposition 3.10. Let S be a symmetric relation in a Hilbert space H. Then the following statements are equivalent:
(i) S is maximally nondensely defined inH;
(ii) S11in(3.1)is an essentially selfadjoint operator indom S, i.e., the semidefect indices of S are equal to(0,0);
(iii) S∞is essentially selfadjoint;
(iv) ran(S−λ)+mulS∗is dense inHfor some (and hence for all)λinC+and for some (and hence for all)λ∈C−;
(v) domS11∗∗=domS∩domS∗;
(vi) PλmulS∗is dense inker(S∗−λ)for some (and hence for all)λ ∈C+and for some (and hence for all)λ∈C−.
Proof. (i)⇔(ii) This follows fromLemma 3.2.
(ii)⇔(iii) This equivalence follows directly from(3.24).
(iii)⇔(iv) This equivalence follows from(3.19).
(ii) ⇔(v) By Lemma 3.7 the equality (S∞)∗∗ = (S∞)∗ can be rewritten as domS11∗∗ = domS∩domS∗, since domS11∗∗⊂domS∗; cf.(3.24).
(iv)⇔(vi) Consider the subspacesM=mulS∗andN =ran(S− ¯λ)withλ∈ C\R. By Lemma 2.2the sum ran(S− ¯λ)+mulS∗ is dense inHif and only if PλmulS∗ is dense in N⊥=ker(S∗−λ).
By combiningLemma 3.8andProposition 3.10, one obtains the following characterizations forS∞to be selfadjoint. The equivalence of the statements (i) and (iv) goes back to [8].
Proposition 3.11. Let S be a symmetric relation in a Hilbert space H. Then the following statements are equivalent:
(i) S∞is selfadjoint;
(ii) S11in(3.1)is a selfadjoint operator indomS;
(iii) ran(S−λ)+mulS∗ =Hfor some (and hence for all)λinC+and for some (and hence for all)λ∈C−;
(iv) domS =dom S∩domS∗;
(v) S is maximally nondensely defined inHand in additiondomS =domS∗∗anddomS∗ = domS∗+dom S.
If the symmetric relation S is closed, then each of the statements(i)–(iv)is also equivalent to the following statement:
(vi) PλmulS∗=ker(S∗−λ)for some (and hence for all)λ∈C+and for some (and hence for all)λ∈C−.
Proof. The equivalence of (i), (ii), (iii), (v), and (vi) is obtained from Lemma 3.8 and Proposition 3.10. The equivalence (i)⇔(iv) follows fromLemma 3.7.
3.3. Characterizations via intermediate extensions
The following theorem gives necessary and sufficient conditions forS∞to be selfadjoint by means of a symmetric extensionT ofSwhich is transversal toS∞.
Theorem 3.12. Let S be a closed symmetric relation in a Hilbert spaceH. Then the following statements are equivalent:
(i) S∞is selfadjoint;
(ii) there exists a closed symmetric extension T of S such that
S∗=T+S∞; (3.25)
(iii) there exists a closed symmetric extension T of S such that
S∗=T+({0} ×mulS∗); (3.26)
(iv) there exists a closed symmetric extension T of S such that
domS∗⊂domT; (3.27)
(v) there exists a closed symmetric extension T of S such that
ker(S∗−λ)⊂domT, ker(S∗− ¯λ)⊂domT, (3.28) for someλ∈C\R.
If a closed symmetric extension T of S satisfies one of (3.25),(3.26),(3.27), or (3.28), then it automatically satisfies the other three of them. Any such closed symmetric relation T is automatically selfadjoint withmulT = mulS; and the selfadjoint extensions T and S∞ are transversal with respect to S:S∗=T+S∞.
Proof. (i)⇒(ii) In fact there exists a selfadjoint extensionT ofS, which is transversal toS∞; cf. [12].
(ii)⇔(iii) This equivalence is obvious.
(iii)⇒(iv) The identity(3.26)implies that domS∗=domT.
(iv)⇒(iii) Observe thatT+({0} ×mulS∗)⊂ S∗for any symmetric extensionT ofS. For the converse inclusion, let{f, f′} ∈S∗. Then f ∈domS∗=domT, and there exists an element h∈Hsuch that{f,h} ∈T ⊂S∗. Hence{0, f′−h} = {f, f′} − {f,h} ∈ S∗, which shows that {f,f′} ∈T+({0} ×mulS∗). ThereforeS∗⊂T +({0} ×mulS∗)and (iii) follows.
(iii)⇒(i) SinceSis closed the identity(3.26)implies the identity
S =T∗∩(dom S⊕H). (3.29)
Now let{f, f′} ∈ (S∞)∗, so that byLemma 3.7{f,f′} ∈ S∗and f ∈ domS. By assumption f ∈ domT ⊂ domT∗ and it follows from(3.29) that f ∈ domS. Hence {f, f′} ∈ S∞ by Lemma 3.7. ThereforeS∞is selfadjoint.
(iv)⇔(v) If(3.27)holds, then certainly(3.28)holds for allλ∈C\R, since ker(S∗−λ)⊂ domS∗ for allλ ∈ C\R. Conversely, assume that(3.28)holds for λ ∈ C\R. Recall von Neumann’s decomposition(2.6)ofS∗, which shows that (withSbeing closed)
domS∗⊂domS+ker(S∗−λ)+ker(S∗− ¯λ).
Now domS⊂domT sinceT is an extension ofS. Hence(3.27)follows.
The proof of the equivalence of (ii), (iii), (iv), (v), has already shown that if a closed symmetric extensionT of Ssatisfies one of the four conditions(3.25),(3.26),(3.27), or(3.28), it automatically satisfies the other three conditions too.
Finally, let T be a closed symmetric extension of S which satisfies(3.25),(3.26), (3.27), or (3.28). In fact, assume that T satisfies (3.27). Then domT = domT∗ = domS∗, and it follows that mulT = mulT∗, sinceT is closed. If{f,f′} ∈ T∗, then f ∈ domT∗ = domT
and there exists an element h such that {f,h} ∈ T ⊂ T∗. Hence {0, f′ − h} ∈ T∗ or f′−h∈mulT∗=mulT, which implies that{f, f′} = {f,h} + {0,f′−h} ∈T. ThereforeT is selfadjoint and mulT =(domT∗)⊥=(domS∗)⊥=mulS.
Corollary 3.13. Let S be a closed symmetric relation in a Hilbert spaceH. Assume that S∞is selfadjoint and that T is a selfadjoint extension of S such that T and S∞are transversal with respect to S. Then
Nλ(S∗)= {(T −λ)−1ϕ;ϕ ∈mulS∗}, λ∈C\R. (3.30) Proof. Let(f, λf)∈S∗. Then it follows from(3.26)that
(f, λf)=(h,h′)+(0, ϕ), (h,h′)∈T, ϕ∈mulS∗.
Therefore, one obtains (h,h′) = (f, λf −ϕ) ∈ T and (f,−ϕ) ∈ T −λ, which leads to f = −(T −λ)−1ϕ. This showsNλ(S∗)⊂(T −λ)−1mulS∗.
Conversely, if f = −(T −λ)−1ϕwithϕ∈mulS∗, then (f,−ϕ+λf)=(−(T −λ)−1ϕ,−ϕ−λ(T −λ)−1ϕ)∈T, so that by(3.26)
(f, λf)=(f,−ϕ+λf)+(0, ϕ)∈T+({0} ×mulS∗)=S∗. This shows{(T−λ)−1ϕ;ϕ ∈mulS∗} ⊂Nλ(S∗).
The next theorem contains a further characterization forS∞to be selfadjoint which involves only properties of the domain of some symmetric extension T of S; it can be also seen as a weakening of the criterion (vi) inProposition 3.11.
Theorem 3.14. Let S be a closed symmetric relation in a Hilbert spaceH. Then S∞is selfadjoint if and only if there is a closed symmetric extension T of S satisfying the following properties:
(i) domT =domS∗;
(ii) domS =dom S∩domT∗; (iii) P(domT∗)=P(domT∗).
In this case T is selfadjoint and transversal to S∞:T+S∞=S∗. Moreover, if S∞is selfadjoint then every selfadjoint extension T of S such that T+S∞=S∗admits the properties(i)–(iii).
In particular, if S is a closed symmetric operator, then S∞is selfadjoint if and only if there exists a closed symmetric operator extension T of S such that
(iv) domT =H;
(v) domS=domS∩domT∗; (vi) P(domT∗)=mulS∗.
Proof. (⇒) Assume thatS∞is selfadjoint; then there exists a selfadjoint extensionT ofSsuch that S∗ = T+S∞; cf. Theorem 3.12. Moreover, byTheorem 3.12mulT = mulS, so that domT = domS∗ and (i) follows. Taking adjoints in(3.26)leads to S = T ∩(dom S×H), which gives (ii). Since the right-hand side of(3.26)is closed and mulT +mulS∗ =mulS∗is closed, it follows fromProposition 2.4thatP(domT)=P(domT), which is (iii).
(⇐) LetT be a closed symmetric extension ofS, so thatS ⊂T ⊂T∗⊂S∗and assume that (i), (ii), and (iii) are satisfied. Introduce the linear relation
H =T+({0} ×mulS∗).
ClearlyS ⊂ H ⊂ S∗ and mulT +mulS∗ =mulS∗. By condition (iii) andProposition 2.4it follows thatHis closed. Furthermore, observe that
H∗=T∗∩(domS×H)=S+({0} ×mulT∗),
where the last identity follows from the assumption (ii). Now by the assumption (i) mulT∗ = mulS and, therefore,H∗= Sor, equivalently,S∗ =H. Hence, the assumptions (i)–(iii) imply that S∗ = H = T+({0} × mulS∗), which by part (iii) ofTheorem 3.12 means that S∞
is selfadjoint and, furthermore, by the same theoremT is necessarily selfadjoint and satisfies S∗=T+S∞.
As to the last statement observe that, ifSis a closed operator, thenS∗is densely defined and, therefore, the statements (i)–(iii) reduce to the statements (iv)–(vi), respectively.
Corollary 3.15.Let S be a closed symmetric relation in a Hilbert space H. Then S∞ is selfadjoint if and only if there is a selfadjoint extension T of S satisfying the following properties:
(i) domS=domS∩domT ; (ii) P(domT)=mulS∗⊖mulS∗∗.
Proof. (⇒) By Theorem 3.14there exists a selfadjoint extension T of S with the properties (i)–(iii) stated therein. In particular, (i) and (iii) inTheorem 3.14combined with(3.9)show that P(domT)=P(domS∗)=mulS∗⊖mulS∗∗.
(⇐) LetT be a selfadjoint extension ofSwith the properties (i) and (ii). Here (ii) means that P(domT)= P(domS∗)and hence P(domT)= P(domT)holds. Since kerP = domS ⊂ domT ⊂ domS∗, the equalityP(domT)= P(dom S∗)implies that domT = domS∗. To complete the proof it remains to applyTheorem 3.14.
4. Partially bounded symmetric relations
Let S be a nondensely defined symmetric relation in a Hilbert space H. Then H admits the orthogonal decomposition H = H1 ⊕H2 = domS ⊕mulS∗. Recall from Section 3.1 the definitions of S11 as a densely defined operator in H1 and of S21 as a densely defined relation fromH1toH2, which satisfyS11∗ ⊂S∗andS∗12 ⊂ S∗. Observe that the corresponding inclusions fail to hold for the linear relations S1 = (I − P)S and S2 = P S in H, since mulS∗ ⊂ domS∗1 = domS∗(I − P) and dom S ⊂ domS2∗ = domS∗P, but in general mulS∗̸⊂domS∗and dom S̸⊂domS∗. This motivates the following definition.
Definition 4.1. A symmetric, not necessarily closed, relationSin a Hilbert spaceHis said to be inner or outer bounded if the closure ofS11orS21, respectively, has a closed domain. Moreover, Sis said to be partially bounded if it is inner or outer bounded.
Partially bounded symmetric relations can be characterized as follows.
Proposition 4.2.Let S be a symmetric relation in a Hilbert space H. Then the following statements are equivalent:
(i) S is inner bounded, i.e.,domS11∗∗is closed;
(ii) S11is a bounded symmetric operator indom S;
(iii) domS ⊂domS∗;
(iv) S∞, as well as(S∗∗)∞, is essentially selfadjoint and their closure(S∞)∗∗ =(S∞)∗has a closed domain.
