Selfadjoint extensions of relations whose domain and range are orthogonal

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space ${\mathfrak H}_1$ to a Hilbert space ${\mathfrak H}_2$ are considered in the Hilbert space ${\mathfrak H}_1\oplus{\mathfrak H}_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in ${\mathfrak H}_1 \oplus {\mathfrak H}_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative selfadjoint extensions $A$, in particular the Friedrichs and Kre\u{\i}n-von Neumann extensions, are parametrized via an explicit block formula. In particular, it is shown that $A$ belongs to the class of extremal extensions of $S$ if and only if ${\rm dom\,} A \perp {\rm ran\,} A$. In addition, using asymptotic properties of an associated Weyl function, it is shown that there is a natural correspondence between semibounded selfadjoint extensions of $S$ and semibounded parameters describing them if and only if the operator part of $R$ is bounded.


Introduction
Let R be a closed linear relation from a Hilbert space H 1 to a Hilbert space H 2 . The problem considered here is to construct selfadjoint relations that extend the relation R in the larger Hilbert space H 1 ⊕ H 2 . Then, based on the case that R is a densely defined closed operator, one expects that the block of linear relations is such a selfadjoint relation. Here the diagonal entries stand for the zero operators on H 1 and H 2 , respectively. Likewise, is also a selfadjoint relation that extends R. The entry {0} × H 2 in this matrix is a purely multivalued relation in H 2 . That these block relations are actually selfadjoint extensions of R is based on the idea that the block representation of R, when considered in the larger space Hilbert space H 1 ⊕ H 2 , given by defines a closed symmetric relation in H 1 ⊕ H 2 , and that the block representation of its adjoint is then given by The above observations are completely formal and need to be justified, i.e., one needs to develop a calculus for 2 × 2 blocks of linear relations; see Remark 2.8 and the text above it. It is not difficult to see that the interpretation of the symmetric relation S in (1.3) leads to the following graph representation: It is clear that S has the property dom S ⊥ ran S and one can show that, in fact, every relation with this property is of the form (1.5). The adjoint of S is given by cf. (1.4). By choosing an appropriate boundary triplet {G, Γ 0 , Γ 1 } all selfadjoint extensions A Θ of S in H can be parametrized by selfadjoint relations Θ in the parameter space G, via A Θ = ker (Γ 1 − ΘΓ 0 ). The selfadjoint extensions in (1.1) and (1.2) correspond to the parameter being the zero operator and the purely multivalued relation, respectively. In particular, the Friedrichs extension S F and the Kreȋn-von Neumann extension S K of S will be determined. In general they are not transversal with respect to S, but they are transversal with respect to S F ∩ S K . This leads to a new boundary triplet by means of which the nonnegative extensions are parametrized by nonnegative relations. On the other hand, by introducing a symmetric extension of S or, loosely speaking, by making the parameter space smaller in an appropriated manner, it will be shown, that depending on whether the operator part R s of R is bounded or not, there is a correspondence between semibounded selfadjoint parameters Θ and semibounded selfadjoint extensions A Θ , or not, respectively.
Here is an overview of the contents of the paper. The notion of a linear block relation is introduced in Section 2. This short treatment is all that is needed in this paper. Section 3 contains a treatment of linear relations whose domain and range are orthogonal. In Section 4 all selfadjoint extensions of S are described by means of an appropriate boundary triplet for S * . A brief intermezzo about nonnegative selfadjoint extensions is given in Section 5. The Friedrichs and Kreȋn-von Neumann extensions and related boundary triplets are studied in Section 6; see Proposition 6.6. A simple description of all nonnegative selfadjoint extensions of S is given in Theorem 6.8 and there is a characterization of all extremal extensions of S in Corollary 6.3. The semibounded extensions of a certain symmetric extension of S are studied in Section 7 by means of the asymptotic behavior of an associated Weyl function. This leads to the alternative mentioned above; see Theorem 7.5.
Blocks of linear relations are built on the treatment of columns and rows of linear relations in [13]. For a related general treatment of blocks of linear operators, see [20]; see also [21]. A characterization of linear relations as block relations will be given later elsewhere; cf. [18]. Note that in the operator case the block in (1.5) was mentioned by Coddington in [6] in connection with a paper of Hestenes [16], who considered selfadjoint operator extensions of arbitrary closed linear operators. For more information in this case, see [19]. The introduction of the corresponding symmetric relation in (1.5), with R being a linear relation, goes back to [6]. The present paper may be seen as a special case of a general completion problem, namely to complete the following block of relations 2. Linear relations with a block structure Before formally introducing blocks of linear relations, here is a brief review of the notions of column and row for pairs of linear relations; cf. [13]. Let H, K, H i , and K i , i = 1, 2, be Hilbert spaces. Let A be a linear relation from H to K 1 and let B be a linear relation from H to K 2 . Then the column col (A ; B) of A and B as a relation from H to Observe that dom col (A ; Next let C be a linear relation from H 1 to K and let D be a linear relation from H 2 to K. Then the row (C; D) of C and D as a relation from H 1 ⊕ H 2 to H is defined by The row of C and D resembles a componentwise sum of linear relations once the domain spaces of C and D are combined in the above way. Observe that ran (C ; D) = ran C + ran D, mul (C ; D) = mul C + mul D.
The following proposition goes back to [13], where one can also find a simple proof. It may be helpful to mention that the definition of an adjoint relation depends on the Hilbert spaces in which the original relation is considered. Thus in each of the following statements one should make sure what Hilbert spaces are involved. (ii) The row of C and D satisfies There are more situations when equality prevails in (i). For instance, if M is a linear subspace in K 2 , and B = H × M one sees by a direct argument that Recall that the domain of col (A; B) is given by dom A ∩ dom B. Hence, if M is a linear subspace in K 2 and B = {0} × M, then it follows that A direct argument then shows that with equality if and only if dom A * × H = A * . Thus, in general, there is no equality in (i). For later use, observe that The relation E is called the block relation corresponding to the block [E ij ].
Forming the row of the two columns in (2.7) by means of (2.3) gives which is the natural way to think of the block relation E. Observe that in the case where all of the relations E ij are everywhere defined bounded linear operators, the block relation E in (2.7) is the usual block operator. It easily follows from the representation and that mul E = (mul E 11 + mul E 12 ) ⊕ (mul E 21 + mul E 22 ).
These two properties distinguish linear block relations among all relations in H.
In Definition 2.3 of a block relation one takes the row of two columns in the block (2.6). In the next lemma it is shown that one obtains the same block relation when taking the column of the two rows in the block (2.6). (2.6). Then

Lemma 2.4. Let [E ij ] be a block as in
Proof. The definition of a column in (2.1) shows that Recall that by the definition of a row in (2.3) one has {f, γ 1 } ∈ (E 11 ; E 12 ) if and only if and, similarly, {f, Combining these facts, one sees that {f, γ 1 } ∈ (E 11 ; E 12 ) and {f, This shows the identity thanks to (2.8). Let . By Lemma 2.4, one sees that

22
, which completes the proof.
As to equality in (2.9), there are the following sufficient conditions; cf. Proposition 2.1 and the identities in (2.4) and (2.5).
Moreover, if N 1 = M ⊥ 1 and N 2 = M ⊥ 2 , then the relation (2.10) is selfadjoint. Proof. The identity (2.10) follows directly from Definition 2.3; see (2.8). The second statement is clear from (2.10), since one sees by a direct argument that for any closed subspace L of a Hilbert space H the linear relation L ⊕ L ⊥ is selfadjoint in H.
Here the notation (M 1 ⊕ M 2 ) ⊤ is a shortcut for the vector notation As a consequence of the above observations, one sees that the block relations ( 3. Linear relations whose domain and range are orthogonal Let S be a linear relation in a Hilbert space H. The interest will be in the rather special case that dom S ⊥ ran S. Clearly, if S has this property, then the same is true for the inverse relation S −1 . Note that the orthogonality condition is always satisfied when either dom S = {0} or ran S = {0}. Here the orthogonality property will be characterized in two different ways. Recall that the numerical range W(S) of a linear relation S in H is defined by It is clear that all eigenvalues in C of S belong to its numerical range W(S). Moreover, for linear relations the numerical range is a convex set; see [15,Proposition 2.18]. Clearly, the numerical range of the inverse of S is given by Here is the first characterization.

Proof. (i) ⇒ (ii) This implication is clear from the definition of W(S).
(ii) ⇒ (i) To prove this reverse implication the following modification of polarization identity is needed: for all {f 1 , g 1 }, {f 2 , g 2 } ∈ S one has Now assume that f 1 ∈ dom S and g 2 ∈ ran S. Then {f 1 , g 1 }, {f 2 , g 2 } ∈ S for some g 1 , f 2 ∈ H. Hence if (ii) holds, then the left-hand side of (3.1) shows that (g 1 , f 2 ) = 0 and thus dom S ⊥ ran S.
Thus, if dom S ⊥ ran S, then it is clear that the relation S is symmetric and that only λ = 0 can be an eigenvalue of S. In fact, the orthogonality property implies that S is semibounded; for instance, S is semibounded from below with lower bound m(S) = 0.
The following result is a characterization of the linear relation in (1.3) and (1.5): it shows that one can express the results in terms of R or S. (i) dom S ⊥ ran S; (ii) H = H 1 ⊕ H 2 and there exists a linear relation R from H 1 to H 2 , such that It follows that S is of the form (1.5). Of course, the choice dom S ⊂ H 1 and ran S ∈ H 2 is arbitrary: one may also interchange the spaces which results in taking the inverse of S.
In the rest of the paper the attention is restricted to linear relations in H for which dom S ⊥ ran S or, equivalently, W(S) = {0}. In this case S is of the form (3.2). The elements of R as a linear relation from H 1 to H 2 will be denoted by {f 1 , f 2 }, but frequently, depending on the situation, also in vector notation by The adjoint R * is a closed linear relation from and J stands for the flip-flop operator J{ϕ, ψ} = {ψ, −ϕ}.

A boundary triplet generated by a closed linear relation
Let S be a closed linear relation in a Hilbert space H for which dom S ⊥ ran S. Then H = H 1 ⊕H 2 and there exists a closed linear relation R from H 1 to H 2 such that S is given by (3.2). In order to describe the selfadjoint extensions of S in H a suitable boundary triplet will be chosen for S * . A first step is the determination of the adjoint S * of S below.
Lemma 4.1. Let R be a closed linear relation from H 1 to H 2 and let S be the closed symmetric relation defined in (3.2). Then Proof. The assertion follows immediately from the identity This identity shows that the right-hand side of(4.1) is contained in the adjoint S * , as (k 1 , f 1 ) − (h 2 , g 2 ) = 0 for all {f 1 , g 2 } ∈ R and {h 2 , k 1 } ∈ R * . The adjoint relation S * is contained in the right-hand side of (4.1) as ( For λ ∈ C the eigenspace associated with (4.1) is given by and, hence, with N λ (S * ) = ker (S * − λ), one has Likewise, the multivalued part of S * is given by The particular form of S * in (4.1) leads to a "natural" boundary triplet for S * ; cf. [5], [10]. For this, one needs to define a parameter space G, and it turns out that It is useful to observe that for {h 1 , h 2 } ∈ G there are the following trivial equivalences: and, likewise Let Q be the orthogonal projection from H 1 ⊕ H 2 onto G.
is an element in S * and define Then Γ 0 and Γ 1 are mappings from S * onto G and {G, Γ 0 , Γ 1 } is a boundary triplet for the relation S * .
Proof. Observe for the element in (4.3) that {h 2 , k 1 } ∈ R * by definition, so that by (4.2) one concludes that Note that Γ 0 and Γ 1 map S * into G. Therefore, for general elements in S * of the form one has the Green identity Thus the abstract Green identity holds with the mappings Γ 0 and Γ 1 in (4.4).
It is clear from the definition of S * that the mapping Γ 0 is onto G. Furthermore, in the definition of S * the elements h 1 ∈ H 1 and k 2 ∈ H 2 are arbitrary; in particular one can choose them as an arbitrary pair in G = N −1 (S * ). Hence, the joint mapping The boundary triplet in (4.4) determines a pair of selfadjoint extensions of S. In particular, H = ker Γ 0 is a selfadjoint extension of S given by and m(H) = 0. It is clear that H is a singular relation as cf. [14]. Note that H coincides with the block relation (1.2). Clearly, the spectrum of H consists only of the eigenvalue 0 ∈ σ p (H), so that ρ(H) = C \ {0}. Note that for λ = 0, it follows from the identity together with (4.1), (4.5), and (4.2), that It is straightforward to see that for ϕ 1 ∈ H 1 and ϕ 2 ∈ H 2 one has These preparations lead to the descriptions for the γ-field and the Weyl function corresponding to the boundary triplet in (4.4).
Theorem 4.3. Let R be a closed linear relation from H 1 to H 2 and let S be the symmetric relation defined in (3.2). Let Q be the orthogonal projection from Let the boundary triplet {G, Γ 0 , Γ 1 } be given by (4.4). Then the corresponding γ-field and Weyl function are given by Proof. Recall that for any λ ∈ C one has that Hence, for the elements in N λ (S * ) it follows from (4.4) that Therefore, by definition, the graph of the Weyl function M is given by or, equivalently, replacing −λh 1 by h 1 , Likewise, by definition, the graph of the γ-field is given by or, equivalently, replacing −λh 1 by h 1 , This completes the proof.
The structure of the Weyl function M in (4.6) gives the following result immediately.
In particular, the identity holds for λ < 0, so that λ → M (λ) is a nondecreasing function on (−∞, 0). The limits M (−∞) and M (0) exist in the strong resolvent sense. Their particular form can be found via the asymptotic behavior of M near λ = −∞ and near λ = 0.
The boundary triplet in Theorem 4.2 can be used to parametrize all selfadjoint extensions of S in (3.2). In fact, the selfadjoint extensions A of S are in one-to-one correspondence with the selfadjoint relations Θ in G, via i.e., in other words In particular, the relation Θ = {0}×G is selfadjoint in G and corresponds to the selfadjoint extension H = ker Γ 0 in (4.5). Likewise, the relation Θ = G × {0}, i.e., Θ = 0, is selfadjoint in G and corresponds to the selfadjoint extension given by whose block representation is given by (1.1); cf. (2.11). In general, the relation K is not semibounded, since (k 2 , h 2 ) = (h 1 , k 1 ) implies which, in general, has no fixed sign. It is clear from (3.2), (4.1), (4.5), and (4.9), that the selfadjoint extensions H and K are transversal, i.e., which, of course, agrees with the identities H = ker Γ 0 and K = ker Γ 1 ; cf. [10], [5].

On nonnegative selfadjoint extensions of nonnegative relations
Let S be nonnegative relation in a Hilbert space H, in other words, (g, f ) ≥ 0 for all {f, g} ∈ S. Such a relation S determines a nonnegative form s on the domain The form s is closable, i.e., its closure s is a closed nonnegative form. On the other hand, if t is a closed nonnegative form in a Hilbert space H, then the first representation theorem asserts that there is a unique nonnegative selfadjoint relation H in H such that t is the closure of the nonnegative form determined by H. This one-to-one correspondence between closed nonnegative forms and nonnegative selfadjoint relations in H is indicated by t = t H . More precisely, t = t Hs , where H s is the selfadjoint operator part of H and mul H = H ⊖ dom t.
If S is a nonnegative relation, then the closure of s is a closed nonnegative form t SF that corresponds to a nonnegative selfadjoint extension S F of S, namely the Friedrichs extension of S. Note that in the case that S is selfadjoint, its so-called Friedrichs extension coincides with S. In general, the Friedrichs extension S F of S can be obtained by Since S is nonnegative, so is S −1 . Therefore, also is a nonnegative selfadjoint extension of S, the so-called Kreȋn-von Neumann extension. Thanks to (5.1) (with S replaced by S −1 ) and (5.2), the Kreȋn-von Neumann extension S K of S can be obtained by The Friedrichs extension and the Kreȋn-von Neumann extension are extreme extensions in the following sense. If A is nonnegative selfadjoint extension of S, then S K ≤ A ≤ S K , or, equivalently, Conversely, if A is a nonnegative selfadjoint relation that satisfies (5.4), then A is an extension, not only of S, but also of the closed symmetric relation S 0 = S F ∩ S K of S, that is S 0 ⊂ A; cf. [5,Theorem 5.4.6]. Consequently, the nonnegative selfadjoint extensions of S and S 0 coincide. Equivalent to the inequalities in (5.4) is that the corresponding forms satisfy cf. [5], where the last inequality actually means t SF ⊂ t A . A nonnegative selfadjoint extension A of S is said to be extremal if It is known that a nonnegative selfadjoint extension A of S is extremal if and only if cf. [3]. For various equivalent conditions for extremality of A, see also [2], [4], and further references in these papers. By the above definition, which uses the inclusion in t SK of the associated closed forms, it is clear that the extremal extensions of S are at the same time also extremal extensions of S 0 and, vice versa.
The case of present interest is where the numerical range of the symmetric relation S in H is trivial: W(S) = {0}; see Section 3. Then the form s determined by S is trivial by Lemma In particular, the form topology coincides with the Hilbert space topology. Then the closure t SF of t S satisfies Therefore, the Friedrichs extension S F of S is given by Proof.
The assumption about S K shows that dom S K ⊥ ran S K . Hence the closed form t SK corresponding to S K is the zero form on the closed domain dom S K . (i) ⇒ (ii) Let A be an extremal extension of S. Then by (5.5) one has t A ⊂ t SK . Hence t A is the zero form on dom t A . In particular, it follows that W(A) = {0}.
(ii) ⇒ (i) Assume that W(A) = {0}, so that the closed form generated by A is the zero form on its necessarily closed domain. By the inequality S K ≤ A one has dom t A ⊂ dom t SK and hence as a zero form t A is a closed restriction of the form t SK , i.e., it satisfies (5.5). Hence A is an extremal extension of S.

Explicit description of all nonnegative selfadjoint extensions
This section contains formulas for the Friedrichs and Kreȋn-von Neumann extensions of S in (3.2). As, in general, they are not transversal as extensions of S, the closed symmetric extension S F ∩ S K of S will be used as the underlying symmetric extension for an alternative boundary triplet. First, the Friedrichs extension S F of S will be determined. Lemma 6.1. Let R be a closed linear relation from H 1 to H 2 and let S be the relation defined in (3.2). Then the Friedrichs extension S F of S is given by Proof. Observe from the definition of S in (3.2) that W(S) = {0} and that Then, thanks to (5.6), one sees that Hence, it follows from (4.1) that (6.1) holds.
Next, the Kreȋn-von Neumann extension S K will be determined in a similar way.
Lemma 6.2. Let R be a closed linear relation from H 1 to H 2 and let S be the relation defined in (3.2). Then the Kreȋn-von Neumann extension S K of S is given by Proof. Observe from the definition of S in (3.2) that W(S −1 ) = {0} and Then, thanks to (5.7), one sees that Hence, it follows from (4.1) that (6.2) holds.
It is clear from Lemma 6.2 that dom S K ⊥ ran S K or, equivalently, W(S K ) = {0}; see Lemma 3.1. Hence from Lemma 5.1 one obtains the following characterization for extremal extensions of S.
cf. Remark 2.8, and, likewise, It follows from the above representations (6.1) and (6.2) that the nonnegative selfadjoint extensions S F and S K of S satisfy Thus S F and S K are disjoint if and only if the relation R is singular. In the opposite case, S F and S K are not disjoint and so not transversal. Now introduce the following symmetric extension of S: Then, by definition, S F and S K are disjoint as selfadjoint extensions of S 0 . It is known that the nonnegative selfadjoint extensions of S and S 0 coincide; cf. Section 6. The following lemma shows that S F and S K are transversal extensions of S 0 .
Lemma 6.5. The adjoint of the symmetric relation S 0 in (6.4) is given by and it satisfies the equality S * 0 = S F + S K . Proof. The description of S * 0 is obtained from (6.4), e.g., by means of the equality S * 0 = JS ⊥ 0 , which shows that 3) and (3.4). The equality S * 0 = S F + S K is now clear from the descriptions of S F in (6.1) and S K in (6.2).
According to Corollary 6.4 the equality S F = S K holds precisely when the subspace In what follows it is assumed that G 0 = {0} and all nonnegative selfadjoint extensions are described. Observe, that G 0 ⊂ G = N −1 (S * ); see (4.2). First notice that for λ ∈ C the eigenspace associated with (6.5) is given by In particular, for λ = 0 the eigenspace N λ (S * 0 ) = ker (S * 0 − λ) has the form Hence, N λ (S * 0 ) = G 0 ⊂ G for all λ = 0. Let Q 0 be the orthogonal projection from H 1 ⊕ H 2 onto G 0 , i.e., Q 0 = P mul R * × P ker R * , where P mul R * is the orthogonal projection from H 1 onto mul R * and where P ker R * is the orthogonal projection from H 2 onto ker R * .
In order to describe all nonnegative selfadjoint extensions of S 0 , it is convenient to construct a boundary triplet {G 0 , Γ 0 0 , Γ 0 1 } for S * 0 such that S F = ker Γ 0 0 and S K = ker Γ 0 1 . Such boundary triplets were introduced and studied by Arlinskiȋ in [1] as a special case of so-called positive boundary triplets (also called positive boundary value spaces) which were introduced earlier by Kochubei [17] and used for describing nonnegative selfadjoint extensions of a nonnegative operator S in the case when 0 is a regular type point of S. The general case was treated also in [7]. A boundary triplet with ker Γ 0 0 = S F and ker Γ 0 1 = S K from [1] is often called a basic (positive) boundary triplet (cf. [4], [5]). Such a boundary triplet is convenient, since all nonnegative selfadjoint extensions of S 0 can be parametrized simply by means of nonnegative selfadjoint relations Θ in the (boundary) space G 0 (cf. Theorem 6.8 below). Proposition 6.6. Let the symmetric relation S 0 be defined by (6.4) with the adjoint (6.5). Let Q 0 be the orthogonal projection from H 1 ⊕ H 2 onto G 0 . Then for Then {G 0 , Γ 0 0 , Γ 0 1 } is a boundary triplet for the relation S * 0 . Furthermore, one has ker Γ 0 0 = S F and ker Γ 0 Proof. For general elements in S * 0 of the form with k 1 , g 1 ∈ mul R * and h 2 , f 2 ∈ ker R * one has the Green identity Thus the abstract Green identity holds with the mappings Γ 0 0 and Γ 0 1 in (6.9). Furthermore, in the definition of S * 0 the elements h 1 ∈ H 1 and h 2 ∈ ker R * are arbitrary and independent from the choice of the elements k 1 ∈ mul R * and k 2 ∈ H 2 . Hence, the pair of mappings (Γ 0 0 , Γ 0 1 ) takes S * 0 onto G 0 × G 0 . Consequently, {G 0 , Γ 0 0 , Γ 0 1 } is a boundary triplet for S * 0 . The identities ker Γ 0 0 = S F and ker Γ 0 1 = S K follow from the definitions in (6.9) and the descriptions of S F in (6.1) and S K in (6.2), respectively. The next result gives the γ-field and the Weyl function corresponding to the boundary triplet {G 0 , Γ 0 0 , Γ 0 1 }.
Proposition 6.7. Let the boundary triplet {G 0 , Γ 0 0 , Γ 0 1 } for S * 0 be as defined in Proposition 6.6. Then the corresponding γ-field and Weyl function are given by Proof. Recall from (6.7) that for any λ = 0 one has that Thus, for the elements in N λ (S * 0 ) it follows from (6.9) and the equality N λ (S * 0 ) = G 0 , λ = 0, in (6.8) that Therefore, by definition, the graph of the Weyl function M 0 is given by i.e., M 0 (λ) = λI G0 . Likewise, by definition, the graph of the γ-field is given by It is possible to describe all nonnegative selfadjoint extensions of S in an explicit form by means of suitable block relation formulas. For this purpose, first notice that Hence S 0 can be decomposed via its operator part (S 0 ) op as follows (6.10) where (S 0 ) mul = {0} × mul S 0 is a selfadjoint relation in ran R which appears as an orthogonal selfadjoint part in the adjoint of S 0 as well as in every selfadjoint extension of S 0 in H 1 ⊕ H 2 . Therefore, it suffices to consider the selfadjoint extensions of the operator part (S 0 ) op in the closed subspace Observe that The adjoint of (S 0 ) op in H 0 is given by see (6.6). It is natural to decompose H 0 as follows Now the following result is obtained from Proposition 6.6 after restricting the mappings Γ 0 0 and Γ 0 1 therein to ((S 0 ) op ) * ; for simplicity the same notation is kept here for these two restrictions; see [5, Remark 2.3.10].
Theorem 6.8. Let the symmetric relation (S 0 ) op be the operator part of S 0 in the subspace H 0 = H 1 ⊕ ker R * with the adjoint (6.11). Let Q 0 0 be the orthogonal projection from H 0 onto G 0 . Then for an element Then {G 0 , Γ 0 0 , Γ 0 1 } is a boundary triplet for the adjoint ((S 0 ) op ) * . Furthermore, the (nonnegative) selfadjoint extensions S Θ of (S 0 ) op in H 0 are in one-to-one correspondence with the (nonnegative) selfadjoint relations Θ in G 0 via where the decomposition is according to In particular, the extremal extensions S Θ of (S 0 ) op are in one-to-one correspondence with the closed subspaces L ⊂ G 0 via Θ = L × (G 0 ⊖ L).
Next observe that since (6.11), one has the following orthogonal componentwise decomposition: ( Therefore, by decomposing f ∈ ((S 0 ) op ) * according to this decomposition in the form Hence, the pair of mappings (Γ 0 0 , Γ 0 1 ) act as the identity mapping on the component G 0 × G 0 and vanishes on the other component (S 0 ) op of ((S 0 ) op ) * . This proves the explicit block formula (6.13) for the selfadjoint extensions of (S 0 ) op .
Due to (6.13), the Kreȋn extension of (S 0 ) op corresponds to Θ = H 0 × {0}. The corresponding form t K is just the zero form on the domain dom t K = H 0 . Since extremal extensions are the nonnegative selfadjoint extensions whose associated closed forms are restrictions of the form t K , they are zero forms on the closed subspaces dom S 0 ⊕ L, where L ⊂ G 0 . This clearly implies the formula for the selfadjoint relations associated to such closed forms and completes the proof.
Note that the (nonnegative) selfadjoint extension S Θ of (S 0 ) op in H 0 can be written as a block relation involving the relation Θ. Such block representations for selfadjoint extensions of a bounded operator can be found in [12,Proposition 5.1], where a different boundary triplet was used; see also [5,Remark 2.4.4]. It is possible to obtain a connection to the boundary triplet in [12] by using the following expression for the adjoint of (S 0 ) op : Notice that the extremal extensions described in Theorem 6.8 correspond to the boundary conditions in G 0 that are determined by the orthogonal projections P L from G 0 onto L; cf. [4,Proposition 7.1]. Recall that orthogonal projections P L are extreme points of the operator interval [0, I G0 ], which also motivates the term "extremal extension" in this situation. There are further descriptions of extremal extensions. In particular, [4,Theorem 8.3] contains a purely analytic description of extremal extensions by means of associated Weyl functions. In the present situation this would lead to the following analytic description: the Weyl functions (of appropriately transformed boundary triplets) of all extremal extensions are of the form:

Semibounded extensions and associated semibounded parameters
In this section semibounded selfadjoint extensions of S are investigated. For this purpose it is convenient to introduce a symmetric extension S of S by reducing the parameter space G slightly, in case the original relation R is not densely defined. The corresponding boundary triplet has a parameter space G ⊂ G and due this restriction the corresponding Weyl function has a specific asymptotic behavior.
Assume that the linear relation R from H 1 to H 2 is closed and let the symmetric relation S in H = H 1 ⊕ H 2 be as in (3.2). Define the linear relation S by Note that dom S ⊥ ran S and that S is a closed symmetric extension of S. It follows from (7.2), together with (4.1), that Observe that matrix representations for S and ( S) * are given by For λ ∈ C the eigenspace associated with (7.3) is given by and, hence, with N λ (( S) * ) = ker (( S) * − λ), one has Since dom S = dom S, one sees that Similar to the situation in Section 4, an eigenspace of ( S) * will play a special role: It is straightforward to see that (cf. (3.3), (3.4)) (7.7) Proposition 7.1. Let R be a closed linear relation from H 1 to H 2 , let S be defined by (7.1) with adjoint (7.3), and let Q be the orthogonal projection from H 1 ⊕ H 2 onto G in (7.6). With an element Then { G, Γ 0 , Γ 1 } is a boundary triplet for ( S) * such that where S F is given by (6.1), and where K is given by (4.9). Moreover, the corresponding Weyl function M (λ) ∈ B( G) is given by Proof. The fact that { G, Γ 0 , Γ 1 } is a boundary triplet for ( S) * can be proved as in Theorem 4.2. To get the formula for the Weyl function M (λ) apply (7.9) to the elements in (7.5) to obtain Here the first entry belongs to G due to {h 2 , λh 1 } ∈ (R * ) s and this leads to (7.12) as in the proof of Theorem 4.3.
To see the identity (7.10), note that the element in (7.8) belongs to ker Γ 0 if and only if It follows from (7.7) that this is the case precisely if h 2 = 0 and k 1 ∈ mul R * , and, consequently, one sees from (7.3) that Comparison with Lemma 6.1 shows that this extension equals the Friedrichs extension S F of S. Likewise, to see the identity (7.11), note that the element in (7.8) belongs to ker Γ 1 if and only if Thanks to (7.7), this is the case precisely if and this equivalence confirms (7.11). As to (7.13) it suffices to check the implication (⇒). By assumption, there exists an element ϕ ∈ mul R * , such that In particular, h 1 + ϕ ∈ dom R, while by definition h 1 ∈ dom R (cf. (7.8)). Thus ϕ ∈ dom R which, together with ϕ ∈ mul R * , implies that ϕ = 0.
Next the Friedrichs and Kreȋn-von Neumann extensions of S will be determined via (5.1) and (5.3).
Lemma 7.2. Let R be a closed linear relation from H 1 to H 2 and let S be the relation defined in (7.1). The Friedrichs extension S F of S is given by Proof. Observe from the definition of S in (7.1) that W( S) = {0} and that Then, thanks to (5.6), one sees that Hence, it follows from (7.3) that (7.14) holds.
Lemma 7.3. Let R be a closed linear relation from H 1 to H 2 and let S be the relation defined in (7.1). The Kreȋn-von Neumann extension S K of S is given by Proof. Observe that W( S −1 ) = {0} and that ran S = (mul R * ⊕ ran R) ⊤ .
Notice that dom S K ⊥ ran S K , so that W( S K ) = {0} and thus A = A * ≥ 0 is an extremal extension of S if and only if W( A) = {0}; see Lemma 5.1.
Recall that ker Γ 0 in Theorem 4.2 is the nonnegative selfadjoint extension H as given in (4.5), while ker Γ 0 in Proposition 7.1 is the Friedrichs extension of S and S. In particular, H ≤ S F and here equality H = S F holds if and only if R is densely defined in H 1 or, equivalently, R * is an operator from H 2 to H 1 . In this case S = S and the boundary triplet in Proposition 7.1 coincides with the one in Theorem 4.2.
For the block representations of the Friedrichs and Kreȋn-von Neumann extensions, note that in terms of block representations one has S F = S F as given in (6.3). It follows from (7.15) and Corollary 2.7 that cf. Remark 2.8 and (7.4).
Observe that the Weyl function M (λ) ∈ B(G) in Theorem 4.3 has the following limit behavior: which is possible when h 1 ∈ mul R * . The Weyl function M (λ) ∈ B( G) in Proposition 7.1 admits the same form as the Weyl function M (λ) ∈ B(G) in Theorem 4.3, but acts in the smaller space G ⊂ G; cf.(4.2), (7.6). In fact, M (λ) is a compression of M (λ) to the subspace G. Hence, as in Corollary 4.4, M (λ) satisfies the following weak identity: where λ ∈ C \ {0}. This leads to an interesting limit result. In fact, it is known that the limit property (7.17) of the Weyl function characterizes ker Γ 0 as the Friedrichs extension; see e.g. [ Now assume that Θ is semibounded from below with lower bound γ ∈ R. Then observe that x < min {−1, −M 2 } and 1 + x M 2 + 1 < γ ⇒ M (x) ≤ γ I ≤ Θ, which, according to (7.18), leads to x ≤ A Θ . Thus, the selfadjoint extension A Θ is bounded from below and this proves the statement.
(ii) Assume that (R * ) s is an unbounded operator. Then for each n ∈ N there exist nontrivial elements {h 2,n , h 1,n } ∈ −(R * ) s such that h 1,n ≥ c n h 2,n , where c n ≥ n. Now it follows from (7.16) that for all x < 0, x + x > 0 and thus for every x < 0 there exists a nontrivial element h ∈ G such that ( M (x)h, h) > 0. Consider a bounded selfadjoint operator Θ in G and assume that A Θ has a lower bound x < 0. Combining the previous reasoning with (7.18) shows that for some h ∈ G Now take Θ = −δI G with δ > 0. Since Θ is a negative definitive operator in G one concludes from (7.21) that the corresponding selfadjoint extension A Θ cannot be semibounded from below. Moreover, here Θ = δ can be made arbitrary small. This completes the proof.
The alternative in Theorem 7.5 can be stated in terms of R, instead of its adjoint, since (R * ) s is a bounded operator precisely when dom R * is closed, which is equivalent to dom R being closed. Thus, the operator part (R * ) s of R * is a bounded (unbounded) operator if and only if the operator part R s of R is a bounded (unbounded) operator. The above proof shows that in case (i) the upper bound of M (x) tends to −∞ as x ↓ −∞, or, in the terminology of [8,9], M (x) tends uniformly to −∞, which is the criterion proved therein for the equivalence: Θ is semibounded ⇔ A Θ is semibounded. It is clear from the proof of (ii) that the upper bound, say ν x , of M (x) satisfies ν x > 0, while M (x) has the weak limit property in (7.17).
It is also possible to describe all nonnegative extensions of the symmetric extension S of S by a treatment similar to the one in Section 6. It follows from (7.14) and (7.15) that S 0 = S F ∩ S K is given by and its adjoint is given by One sees immediately that for all λ ∈ C The details are left to the reader.