Boundary Triplets and Weyl Functions

Green applied to (cid:5) and k shows

Note that a symmetric relation S is densely defined if and only if S * is an operator. In this case the boundary mappings Γ 0 and Γ 1 can be defined on dom S * instead of (the graph of) S * . More precisely, if {G, Γ 0 , Γ 1 } is a boundary triplet for S * , then one defines boundary mappings Γ 0 and Γ 1 on dom S * by the following identifications In the following treatment whenever S is a densely defined operator, boundary mappings defined on S * and on dom S * will be identified in this sense. After this identification (2.1.1) turns into (2.1.2) where f, g ∈ dom S * . This formalism will be used in Chapter 6 and Chapter 8 in the treatment of ordinary and partial differential operators.
The identity (2.1.1) or the identity (2.1.2) is sometimes called the abstract Green identity or the abstract Lagrange identity; in this text mostly the terminology abstract Green identity will be used. This identity has a geometric interpretation which is best expressed in terms of the indefinite inner products ·, · H 2 := J H ·, · H 2 , where J H = J * H = J −1 H ∈ B(H 2 ) and J G = J * G = J −1 G ∈ B(G 2 ); cf. Section 1.8. By means of these inner products, the identity (2.1.1) can be rewritten as f, g H 2 = Γ f, Γ g G 2 (2.1.4) for f = {f, f }, g = {g, g } ∈ S * . Later the scalar products in (2.1.1), (2.1.2), and (2.1.4) will be used without indices H and G, respectively, when there is no danger of confusion. Recall that the adjoint A * of a relation A in H can be written as an orthogonal complement with respect to the inner product [[·, ·]], that is A * = A [[⊥]] ; cf. Section 1.8.
Some elementary but important properties of the boundary mappings are collected in the following proposition.
Clearly, this gives rise to a one-to-one correspondence between all intermediate extensions H of S and all subspaces H of M via H = S ⊕ H , which is expressed in (2.1.5). Moreover, since Γ is an isomorphism it also follows from (2.1.7) that Θ = Γ H = Γ H and hence the closure H of H corresponds to the closure Θ of Θ. This implies via (2.1.5) that A Θ = A Θ .
(iii) Let A Θ be defined by (2.1.5). It will be verified that holds for any relation Θ in G. Note that (2.1.8) is clear in the special case that Θ is an operator, since Γ f = {Γ 0 f, Γ 1 f } ∈ Θ means that ΘΓ 0 f = Γ 1 f . Now assume that Θ is a relation. First the inclusion (⊂) in (2.1.8) will be shown. For this consider f ∈ A Θ . Hence, f ∈ S * and {Γ 0 f, In other words f ∈ ker (Γ 1 −ΘΓ 0 ).
(vi) Let A Θ be an operator. Then clearly S is an operator. Assume that Γ f ∈ Θ for some element f = {0, f } ∈ S * . Then f ∈ A Θ and hence f = 0, so that (2.1.6) holds.
Due to the abstract Green identity (2.1.1), (2.1.2), or (2.1.4) some properties of intermediate extensions are preserved in the corresponding relations in G.
Corollary 2.1.4. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , and let A Θ be the extension of S in H corresponding to the relation Θ in G via (2.1.5). Then the following statements hold: (accumulative) if and only if Θ is dissipative (accumulative); (ii) A Θ is maximal dissipative (maximal accumulative) if and only if Θ is maximal dissipative (maximal accumulative); (iii) A Θ is symmetric if and only if Θ is symmetric; (iv) A Θ is maximal symmetric if and only if Θ is maximal symmetric; (v) A Θ is self-adjoint if and only if Θ is self-adjoint.
(ii) According to Theorem 2.1.3 (v), for any two extensions A Θ and A Θ of S one has A Θ ⊂ A Θ if and only if Θ ⊂ Θ holds. Therefore, if A Θ is maximal dissipative, then Θ is dissipative because of (i) and if Θ is a dissipative extension of Θ in G, then A Θ is a dissipative extension of A Θ , so that A Θ = A Θ . Hence, Θ = Θ and Θ is maximal dissipative. The converse direction is proved in exactly the same way. The statement for maximal accumulative extensions follows analogously.
(iii)-(v) These assertions follow from the previous items, and the fact that a relation is symmetric (self-adjoint) if and only if it is (maximal) dissipative and (maximal) accumulative. Proof. The identity (2.1.9) follows from while the identity (2.1.10) follows from In particular, (2.1.9) together with S = ker Γ shows that A Θ ∩ A Θ = S if and only if Θ ∩ Θ = {0, 0}, while (2.1.9) and (2.1.10) show that A Θ ∩ A Θ = S and A Θ + A Θ = S * if and only if Θ ∩ Θ = {0, 0} and Θ + Θ = G 2 . This completes the proof.
Corollary 2.1.6. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * and assume that dim G < ∞. If A Θ and A Θ are selfadjoint extensions of S which are disjoint, then they are transversal.
Let S be a closed symmetric relation in H and let {G, Γ 0 , Γ 1 } be a boundary triplet for S * . There are two special extensions of S which will be frequently used in the following; they are defined by A 0 := ker Γ 0 and A 1 := ker Γ 1 . (2.1.11) It is clear that A 0 and A 1 are self-adjoint extensions of S, since they correspond to the self-adjoint parameters Θ in G in (2.1.5) given by Θ = {0} × G and Θ = G × {0}, (2.1.12) respectively. Furthermore, the representations in (2.1.12) show that the self-adjoint extensions A 0 and A 1 are transversal; cf. Lemma 2.1.5. Note also in this context that a boundary triplet {G, Γ 0 , Γ 1 } for S * can only exist if the defect numbers of the closed symmetric relation S coincide (since it admits the self-adjoint extensions A 0 and A 1 in (2.1.11)); a more detailed discussion on the existence and uniqueness of boundary triplets will be provided in Section 2.4 and Section 2.5. As S = ker Γ, it follows from von Neumann's decomposition Theorem 1.7.11 that Γ is an isomorphism from N λ (S * ) + N λ (S * ), λ ∈ C \ R, onto G 2 . Due to the definitions in (2.1.11) a similar observation can be made for the components Γ 0 and Γ 1 .
Proof. As A 0 and A 1 are self-adjoint, the direct sum decompositions (2.1.13) hold by Corollary 1.7.5. Since Γ 0 and Γ 1 map S * onto G, and A 0 and A 1 are their respective kernels, it is clear that the restrictions Γ 0 N λ (S * ) and Γ 1 N λ (S * ) are isomorphisms from N λ (S * ) onto G.
In the rest of the text the self-adjoint extension A 0 = ker Γ 0 will often serve as a point of reference due to the corresponding representation {0} × G in the parameter space G. In the next proposition it is shown that A 0 and a given closed extension A Θ are disjoint (transversal) if and only if the parameter Θ is a (bounded everywhere defined) operator.
which is the same as saying that mul Θ = {0} and dom Θ = G. By the closed graph theorem, the last two conditions are equivalent to Θ ∈ B(G).
Let S be a closed symmetric relation in H with equal defect numbers and let H be a self-adjoint extension of S. Later it will be shown that there exists a boundary triplet {G, Γ 0 , Γ 1 } for S * such that ker Γ 0 concides with H; cf. Theorem 2.4.1. Furthermore, it will be shown that for a pair of self-adjoint extensions of S which are transversal, there exists a boundary triplet {G, Γ 0 , Γ 1 } for S * such that ker Γ 0 and ker Γ 1 coincide with this pair; cf. Theorem 2.5.9. The notion of boundary triplet is not unique; in fact, a parametrization of all possible boundary triplets will be provided in Section 2.5.
The following theorem is of a different nature. It can be used to prove that a given relation T is the adjoint of a symmetric relation S. Theorem 2.1.9. Let T be a relation in H, let G be a Hilbert space, and assume that is a linear mapping such that the following conditions are satisfied: Then S := ker Γ is a closed symmetric relation in H such that S * = T and Proof. First note that condition (iii) implies that ker Γ 0 is symmetric. To see this, let f, g ∈ ker Γ 0 . Then, by condition (iii), and hence ker Γ 0 is a symmetric relation in H.
It follows from S = T * that S * = T * * = T . Hence, it remains to show that T is closed. Let ( f n ) be a sequence in T converging to f . It suffices to show that f ∈ T . Let ψ ∈ G 2 and let g ∈ T be such that ψ = J −1 G Γ g (here condition (ii) is being used). Using the continuity of the indefinite inner product [[·, ·]] (see Section 1.8) one obtains This shows that Γ f n is a weak Cauchy sequence in G, hence weakly bounded and thus bounded. It follows that there exists a subsequence, again denoted by Γ f n , which converges weakly to some ϕ ∈ G × G. Now let h ∈ T be such that Γ h = ϕ (again condition (ii) is being used). Choose g ∈ T and let, as above, ψ = J −1 G Γ g, so that (2.1.15) remains valid. Then (2.1.15) implies Therefore, T is closed and it follows that S * = T . By conditions (ii) and (iii) {G, Γ 0 , Γ 1 } is a boundary triplet for S * . Above it was also shown that A 0 = ker Γ 0 .

Boundary value problems
Let S be a closed symmetric relation in H and let {G, Γ 0 , Γ 1 } be a boundary triplet for S * . Due to Theorem 2.1.3 one may think of the intermediate extensions of S being parametrized by the relations in the space G; for this reason the space G will often be called the boundary space or parameter space associated with the boundary triplet. Let Θ be a closed relation in G and let A Θ be the corresponding closed extension of S in H via (2.1.5): (2.2.1) Recall from Section 1.10 that any closed relation Θ in G has a parametric representation of the form Θ = {A, B}, i.e., with some operators A, B ∈ B(E, G) and a Hilbert space E. Likewise, since Θ * is closed, it has a representation of the form with some operators C, D ∈ B(E , G) and a Hilbert space E . Thus, (2.2.3) gives Therefore, it follows that A Θ in (2.2.1) can be written as Proof. By Proposition 1.10.10, condition (2.2.6) implies that the relation Θ is given by Then (2.2.7) follows from (2.1.5).
In the next proposition it will be assumed, in addition, that ρ(Θ) = ∅. The following result is a reformulation of Theorem 1.10.5 and formula (2.1.5).
For μ ∈ ρ(Θ) it follows from (1.10.9) that in Proposition 2.2.2 one can choose In the case that μ ∈ C \ R one may also choose for some, and hence for all μ ∈ C + and for some, and hence for all for some, and hence for all μ ∈ C − ; (iii) A Θ is maximal accumulative if and only if for some, and hence for all μ ∈ C + .
In the case that the representation Θ = {A, B} is chosen so that Θ * = {A * , B * }, the extension A Θ is given by (2.2.8). Now the converse question will be addressed. Let A be a closed extension of S given in terms of boundary conditions. The problem is to determine a corresponding parameter Θ in G such that A = A Θ . Proposition 2.2.4. Let S be a closed symmetric relation in H, and let {G, Γ 0 , Γ 1 } be a boundary triplet for S * . Assume that F is a Hilbert space, M, N ∈ B(G, F), and that, without loss of generality, the space F is minimal: Then A is closed and A = A Θ , where the parameter Θ = {A, B} is given by Proof. First observe that the intermediate extension A in (2.2.11) is closed since M, N ∈ B(G, F). Moreover, A corresponds to the closed relation Θ in G given by Now the assertion follows from Proposition 1.10.7.
Let again Θ be a closed relation in G and let A Θ be the corresponding closed extension in H via (2.1.5). Assume, in addition, that Θ admits an orthogonal decomposition into a (not necessarily densely defined) operator part Θ op acting in the Hilbert space G op = dom Θ * = (mul Θ) ⊥ and a multivalued part Θ mul = {0} × mul Θ in the Hilbert space G mul = mul Θ; cf. Theorem 1.3.16 and the discussion following it.
Recall from Theorem 1.4.11, Theorem 1.5.1, and Theorem 1.6.12 that any closed symmetric, self-adjoint, (maximal) dissipative, or (maximal) accumulative relation Θ in G gives rise to such a decomposition. If P op denotes the orthogonal projection in G onto G op , then the closed extension A Θ in (2.1.5) has the form Note that this abstract boundary condition also requires P op Γ 0 f ∈ dom Θ op .
Definition 2.3.1. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , and let A 0 = ker Γ 0 . Then is called the γ-field associated with the boundary triplet {G, Γ 0 , Γ 1 }.
The main properties of the γ-field will now be discussed.
(ii) Let λ, μ ∈ ρ(A 0 ) and let ϕ ∈ G. Then there exists Moreover, this observation also shows that for some g ∈ H (iii) Fix some μ ∈ ρ(A 0 ). Then it follows from (ii) and the fact that the mapping λ → (A 0 − λ) −1 is a holomorphic operator function with values in B(H) that λ → γ(λ) is a holomorphic operator function on ρ(A 0 ) with values in B(G, H).
In the case where the symmetric relation S is a densely defined symmetric operator and {G, Γ 0 , Γ 1 } is a boundary triplet for S * with boundary mappings Γ 0 and Γ 1 defined on dom S * (see the text below Definition 2.1.1 and (2.1.2)) the formula for the adjoint γ(λ) * of the corresponding γ-field in Proposition 2.3.2 (iv) has the simpler form According to Proposition 2.3.2 (iv), the action of Γ 1 on a general element of A 0 is expressed in terms of the operator γ(λ) * . The form of this action is particularly simple on eigenelements of A 0 .
The definition and properties of the γ-field now give rise to the notion of Weyl function. It is defined, as in the case of the γ-field, for a closed symmetric relation S in terms of the boundary triplet for S * and the eigenspaces of S * . Definition 2.3.4. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , and let A 0 = ker Γ 0 . Then is called the Weyl function associated with the boundary triplet {G, Γ 0 , Γ 1 }.
Here is a simple example of a Weyl function for a trivial symmetric relation S in H. Note that in this example one has G = H, i.e., the corresponding boundary triplet maps onto H × H; this situation is not typical in standard applications; cf. Chapters 6, 7, and 8. Next some elementary properties of the Weyl function are discussed. Recall that the real part and imaginary part of a bounded operator T ∈ B(G) are defined as Re T = 1 2 (T + T * ) and Im T = 1 2i (T − T * ), respectively. Proposition 2.3.6. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , and let A 0 = ker Γ 0 . Then the following statements hold for the corresponding γ-field γ and Weyl function M : holds, and, in particular, the symmetry condition Proof. (i) Let λ ∈ ρ(A 0 ). By Lemma 2.1.7, the restriction of Γ 0 to N λ (S * ) is an isomorphism between N λ (S * ) and G. Hence, the inverse γ(λ) is an isomorphism between G and N λ (S * ), and since the operator Γ 1 : S * → G is continuous by Proposition 2.1.2 (i), it follows from Definition 2.3.4 that M (λ) = Γ 1 γ(λ) ∈ B(G).
In the next corollary it turns out that the Weyl function M is a uniformly strict Nevanlinna function; cf. Definition A.4.1 and Definition A.4.7.

Corollary 2.3.7. The Weyl function M in Definition 2.3.4 is a uniformly strict B(G)-valued Nevanlinna function. Its values
Proof. According to Proposition 2.3.2 (iii), the function λ → γ(λ) is holomorphic on ρ(A 0 ). Hence, it follows from Proposition 2.3.6 (iii) with fixed μ ∈ ρ(A 0 ) that the function λ → M (λ) is holomorphic on ρ(A 0 ) and hence, in particular, on the possibly smaller subset C \ R. Clearly, according to Proposition 2.3.6 (iii) and (iv) one has M (λ) * = M (λ) and (Im λ)(Im M (λ)) ≥ 0 for λ ∈ C \ R, and hence M is a B(G)-valued Nevanlinna function. It follows from Proposition 2.3.6 (iv) that M is uniformly strict. and in the strong graph sense or, equivalently, in the strong resolvent sense on C \ R.
exists. Moreover, for all ϕ ∈ G and y = x Proposition 2.3.6 (iii) shows that If γ y is a lower bound for M (y), then Corollary 1.9.10 implies that there exists a semibounded self-adjoint relation M (b) such that M (x) converges in the strong resolvent sense to M (b) on C \ [γ y , ∞) when x tends to b. According to Corollary 1.9.6 (i), this is equivalent to strong graph convergence of M (x) to M (b).
The same considerations as above show that (−M (y)ϕ, ϕ) ≤ (−M (x)ϕ, ϕ) for a < x < y < b and ϕ ∈ G, and hence −M (x) converges in the strong resolvent sense to a semibounded self-adjoint relation −M (a) on C \ [γ y , ∞) when x tends to a; here γ y is a lower bound for −M (y). This implies that M (x) tends to M (a) in the strong graph sense and in the strong resolvent sense.
It is known that every isolated spectral point of a self-adjoint operator or relation A 0 is an eigenvalue and a pole of first order of the resolvent λ → (A 0 −λ) −1 . As a consequence of Proposition 2.3.6 (v), the isolated singularities of the Weyl function M are poles of first order. This is formulated in the next corollary, which can also be regarded as a simple example of the connection between the properties of the Weyl function M and the spectrum of A 0 . The full connection between these objects is studied in detail in Section 3.5 and Section 3.6.

then M admits a norm convergent Laurent series expansion of the form
In particular, In the next remark it is explained that a self-adjoint part of a symmetric relation has, roughly speaking, no influence on the corresponding boundary triplet, γ-field, and Weyl function. Remark 2.3.10. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , and let A 0 = ker Γ 0 . Assume that H admits an orthogonal decomposition H = H ⊕ H and that S has the orthogonal decomposition where S is a closed symmetric relation in H and A is a self-adjoint relation in H . Then it follows from (2.3.5) and Proposition 1.3.13 that where (S ) * stands for the adjoint of S in the space H . Observe that according to (2.3.6) every element {f, f } ∈ S * has the decomposition Hence, if Γ 0 and Γ 1 denote the restrictions of Γ 0 and Γ 1 to (S ) * , then it is easily seen that {G, Γ 0 , Γ 1 } is a boundary triplet for (S ) * such that Moreover, note that (2.3.6) shows which implies that the Weyl function M and the γ-field γ satisfy

For completeness observe that if H is a closed intermediate extension of S and
Hence, one may discard the self-adjoint part A in H without disturbing the boundary triplet structure.

Existence and construction of boundary triplets
Here the existence and possible constructions of boundary triplets based on decompositions in Section 1.7 are addressed. Recall first from Corollary 1.7.13 that a closed symmetric relation S in a Hilbert space H admits self-adjoint extensions in H if and only if the defect numbers dim N λ (S * ) and dim N μ (S * ) (2.4.1) of S are equal for some, and hence for all λ ∈ C + and some, and hence for all μ ∈ C − . Since any boundary triplet for S * induces two self-adjoint extensions A 0 and A 1 of S it is clear that a boundary triplet can only exist if the defect numbers are equal. It turns out that this condition is also sufficient.
The following main result makes explicit how to construct a boundary triplet in terms of a given self-adjoint extension of a closed symmetric relation S (which exists if and only if the defect numbers in (2.4.1) coincide). The following notation will be used. For μ ∈ C the natural embedding of N μ (S * ) into H is denoted by ι Nμ(S * ) and its adjoint is the orthogonal projection P Nμ(S * ) from H onto N μ (S * ).

Theorem 2.4.1. Let S be a closed symmetric relation in H and assume that H is a self-adjoint extension of S in H. Fix μ ∈ ρ(H) and decompose S * as
Then and the corresponding Weyl function M is given by where g 0 = {g 0 , g 0 } ∈ H and g μ = {g μ , μg μ } ∈ N μ (S * ). Then it follows directly from the decompositions (2.4.3), (2.4.7), and (f 0 , g 0 ) = (f 0 , g 0 ) that (2.4.8) Moreover, it follows from the definition (2.4.4) applied to f and g that (2.4.9) A combination of (2.4.8) and (2.4.9) shows that the abstract Green identity (2.1.1) holds.
It is interesting to see what Theorem 2.4.1 means in the simple case when the underlying symmetric relation is trivial; cf. Example 2.3.5, which is opposite in the sense that there ker Γ 0 = H ×{0} and M (λ) = −(1/λ)I. Again this example is not typical, since in standard applications G = H; cf. Chapters 6, 7, and 8.
Therefore, one has the direct sum decomposition and any f ∈ S * has the corresponding decomposition According to (2.4.4), one sees that and that for every λ ∈ C the resolvent (H − λ) −1 is the zero operator. Hence, the γ-field is given by γ(λ) = I and the Weyl function is given by M (λ) = λI.
There is an addendum to Theorem 2.4.1 when the decomposition (2.4.2) is replaced by a decomposition involving In fact, the following result may be seen as a limit result obtained from (2.4.2) with μ → ∞. The embedding of N ∞ (S * ) = mul S * into H is denoted by ι N∞(S * ) and its adjoint is the orthogonal projection P N∞(S * ) from H onto N ∞ (S * ). The proof of Proposition 2.4.3 is straightforward. Observe that Example 2.3.5 is an illustration of the following proposition.
Proposition 2.4.3. Let S be a closed symmetric operator in H and assume that H is a self-adjoint extension of S which belongs to B(H). Then S * can be decomposed as Then and the corresponding Weyl function M is given by Remark 2.4.4. Let S be a closed symmetric operator in H and let H ∈ B(H) be a self-adjoint extension of S as in Proposition 2.4.3. Then S is bounded and hence dom S is closed. Decompose H = dom S ⊕ N ∞ (S * ) and note that and in a similar way It follows that H 11 = S 11 , H 21 = S 21 , and H * 21 = S * 21 . Relative to the decomposition (2.4.12) of S * , the boundary triplet in (2.4.13) can be written as Let Θ be a closed relation in G = N ∞ (S * ). Then the corresponding extension A Θ of S is given by which can formally be written as Therefore, the extensions of S may be interpreted as solutions of the completion problem posed by the incomplete 2 × 2 operator matrix Theorem 2.4.1 has some variations when the self-adjoint extension H in (2.4.2) is further decomposed; cf. Section 1.7. The most straightforward results are presented in the following corollaries. In the next result the direct sum decomposition from Corollary 1.7.10 (with μ = λ) is used.
Proof. It follows from Corollary 1.7.10 that every f = {f, f } ∈ S * can be written as with {h, h } ∈ S and k ∈ N μ (S * ). The boundary mappings in Theorem 2.4.1 then have the form Γ 0 f = f μ and where h − μh ∈ ran (S − μ) = N μ (S * ) ⊥ and k + μf μ ∈ N μ (S * ) was used in the last step. This shows that the mappings in (2.4.15) form a boundary triplet with the same γ-field and Weyl function as in Theorem 2.4.1.
In Theorem 2.4.1, Proposition 2.4.3, and Corollary 2.4.5 the boundary triplets were based on decompositions of S * in Section 1.7. The following result gives a boundary triplet for a decomposition of S * which is a mixture of the above decompositions.
Corollary 2.4.6. Let S be a closed symmetric relation in H, assume that H is a self-adjoint extension of S in H, and fix μ ∈ C \ R. Every f = {f, f } ∈ S * has the unique decomposition Then define a boundary triplet {N μ (S * ), Γ 0 , Γ 1 } for S * such that H = ker Γ 0 . The corresponding γ-field and Weyl function are given by (2.4.5) and (2.4.6).
Proof. Let f = {f, f } ∈ S * . Then according to (2.4.14) there is the decomposition where h = {h, h } ∈ S, k ∈ N μ (S * ), and ψ ∈ N μ (S * ) are uniquely determined. Define the element ϕ by k = μψ + ϕ, so that ϕ ∈ N μ (S * ) and the right-hand side of the above decomposition can be rewritten as By von Neumann's second formula (see Theorem 1.7.12) one can describe the self-adjoint extension H in Theorem 2.4.1 by means of an isometric operator from N μ (S * ) onto N μ (S * ). This observation also gives rise to the construction of a boundary triplet, where the parameter space is given by N μ (S * ).
Theorem 2.4.7. Let S be a closed symmetric relation in H, let H be a self-adjoint extension of S, and fix some μ ∈ C \ R. Let W be the isometric mapping from and decompose f = {f, f } ∈ S * according to von Neumann's first formula: Then define a boundary triplet {N μ (S * ), Γ 0 , Γ 1 } for S * such that H = ker Γ 0 . The corresponding γ-field and the Weyl function are given by (2.4.5) and (2.4.6).
Proof. Let f = {f, f } ∈ S * be decomposed as in (2.4.18) and let g = {g, g } ∈ S * be decomposed in the analogous form and therefore On the other hand it follows from (2.4.19) that i.e., the abstract Green identity (2.1.1) holds.
Note that the strategy in the proof of Theorem 2.4.7 is different from the strategy in the two previous results. The connection will be sketched now. In Theorem 2.4.7 the isometric mapping W from N μ (S * ) onto N μ (S * ) determines the boundary triplet {N μ (S * ), Γ 0 , Γ 1 } for S * in (2.4.19). The self-adjoint extension H of S determined by W in (2.4.17) then satisfies H = ker Γ 0 . Now apply Theorem 2.4.1 with this particular self-adjoint extension. Hence, f ∈ S * in Theorem 2.4.1 is decomposed in the form (2.4.20) where {f 0 , f 0 } ∈ H and {ϕ μ , μϕ μ } ∈ N μ (S * ). Making use of the decomposition (2.4.17) of H it follows that with {h, h } ∈ S and ψ μ ∈ N μ (S * ). Therefore, f in (2.4.20) is given by shows that the boundary maps Γ 0 in Theorem 2.4.1 and Theorem 2.4.7 coincide. Moreover, as P Nμ(S * ) (h − μh) = 0, the identity

Transformations of boundary triplets
Let S be a closed symmetric relation in H with equal defect numbers. Then S admits self-adjoint extensions in H and each self-adjoint extension gives rise to a boundary triplet as in Theorem 2.4.1. Hence, boundary triplets for S * are not uniquely determined, with the exception of the trivial case S = S * . A complete description of all boundary triplets for S * will be given with the help of block operator matrices that are unitary with respect to the indefinite inner products in Section 1.8; cf. (2.1.3). The transformation properties of the corresponding boundary parameters, γ-fields, and Weyl functions are discussed afterwards.
The main result on the description of all boundary triplets for S * is the following theorem. It describes the transformation of boundary triplets.
Theorem 2.5.1. Let S be a closed symmetric relation in H, assume that {G, Γ 0 , Γ 1 } is a boundary triplet for S * , and let G be a Hilbert space. Then the following statements hold: Therefore, Γ 0 and Γ 1 satisfy the abstract Green identity (2.1.1); cf. (2.1.4). Since W is surjective by Proposition 1.8.2, Γ = WΓ is also surjective thanks to the surjectivity of Γ. It follows that {G , Γ 0 , Γ 1 } is a boundary triplet for S * .
(ii) Assume that {G , Γ 0 , Γ 1 } is a boundary triplet for S * and define a linear relation It follows from Proposition 2.1.2 (ii) that W is an operator. Indeed, if Γ f = 0, then f ∈ S and thus Γ f = 0. For the operator W one has dom W = G × G and ran W = G × G since ran Γ = G × G and ran Γ = G × G .
The transformation of a boundary triplet {G, Γ 0 , Γ 1 } in Theorem 2.5.1 induces a transformation of the closed relations in the parameter space. Assume that W ∈ B(G × G, G × G ) satisfies the identities in (2.5.1) and let {G , Γ 0 , Γ 1 } be the corresponding transformed boundary triplet in (2.5.2). Let Θ be a relation in G and define Θ in G as a Möbius transform of Θ by Proof. Let f ∈ S * . Then the transformation formulas (2.5.2), (2.5.4), and the fact that W is bijective imply Hence, ker (Γ 1 − ΘΓ 0 ) and ker (Γ 1 − Θ Γ 0 ) coincide; cf. Theorem 2.1.3 (iii).
Likewise, the transformation of the boundary triplet leads to a transformation of the γ-field and the Weyl function. Proof. For λ ∈ ρ(A 0 ) and f λ ∈ N λ (S * ) one has If, in addition, λ ∈ ρ(A 0 ) ∩ ρ(A 0 ), then, by Lemma 2.1.7, Γ 0 and Γ 0 are isomorphisms from N λ (S * ) onto G and G , respectively. Hence, it follows from (2.5.7) that W 11 + W 12 M (λ) is an isomorphism from G onto G , and therefore If one applies the orthogonal projection π 1 from H × H onto H × {0} to both sides, then (2.5.5) follows. Similarly, for λ ∈ ρ(A 0 ) ∩ ρ(A 0 ) one finds that which yields (2.5.6).
Next some special transformations of boundary triplets and Weyl functions will be discussed. In the first corollary the boundary mappings are interchanged via a flip-flop, which leads to the Weyl function −M −1 . Proof. The operator

Corollary 2.5.4. Let S be a closed symmetric relation in H and assume that
satisfies both identities in (2.5.1). Now the assertions follow from Theorem 2.5.1 and Proposition 2.5.3.
The second corollary treats the situation in which a bijective operator D dilates the Weyl function M and a self-adjoint operator P produces a shift of the dilated Weyl function D * MD.

respectively.
Proof. It is not difficult to check that the operator satisfies both identities in (2.5.1). Now the assertions follow from Theorem 2.5.1 and Proposition 2.5.3. The next corollary complements Corollary 2.5.5.
A combination of Corollary 2.5.5 with G = G , D = I, P = −Θ, and Corollary 2.5.4 leads to the followings statement.

respectively.
The following statement is also a direct consequence of Corollary 2.5.5.
Proof. Due to Proposition 2.3.6 (iii) it follows from the identity (2.5.11) that and, in particular, Im Q(λ 0 ) = Im M (λ 0 ). Hence, one obtains With the choice D = I and P as above, the result follows from Corollary 2.5.5. Now it will be shown that a pair of transversal self-adjoint extensions induces a boundary triplet {G, Γ 0 , Γ 1 } which determines these extensions via the boundary conditions ker Γ 0 and ker Γ 1 . The following theorem is a consequence of Theorem 2.4.1 and Corollary 2.5.5.
Corollary 2.5.10 is concerned with the existence of the boundary triplet {G, Γ 0 , Γ 1 } with the properties (2.5.13). In fact, it is possible to explicitly construct such a boundary triplet via the choice of an appropriate operator W such that Γ = WΓ; cf. Corollary 2.5.7 for the special case Θ ∈ B(G).
Corollary 2.5.11. Let S be a closed symmetric relation in H and assume that {G, Γ 0 , Γ 1 } is a boundary triplet for S * . Let Θ be a self-adjoint relation in G and choose A, B ∈ B(G) such that Θ = {Aϕ, Bϕ} : ϕ ∈ G and the identities (2.5.14) hold; cf. Corollary 1.10.9. Then {G, is a boundary triplet for S * such that both identities in (2.5.13) hold, that is, Proof. It is not difficult to check that and since −Θ −1 = {{Bϕ, −Aϕ} : ϕ ∈ G}, it follows in the same way that Recall from Proposition 2.5.2 that holds for any closed relation Ξ in G. With Ξ = Θ and Ξ = −Θ −1 one then has respectively. Now (2.1.11)-(2.1.12) imply Assume that the boundary triplets {G, Γ 0 , Γ 1 } and {G, Γ 0 , Γ 1 } are as in Corollary 2.5.11 and let γ and M , and γ and M , be the corresponding γ-fields and Weyl functions, respectively. Then it follows from Proposition 2.5.3 that for all (2.5.18) In the special case where the defect numbers of S are (1, 1) one may choose where A = 0 and B = 1 if s = ∞; this interpretation will be used also in the following. With this choice of A and B the operator in (2.5.16) reduces to the 2 × 2-matrix In this case (2.5.19) and for λ ∈ ρ(A 0 ) ∩ ρ(A 0 ) the corresponding γ-field and Weyl function are given by Now let S be a closed symmetric relation in H and assume that {G, Γ 0 , Γ 1 } is a boundary triplet for S * . Consider a closed symmetric extension S of S with the property S ⊂ A 0 = ker Γ 0 . Then the boundary triplet {G, Γ 0 , Γ 1 } can be restricted to (S ) * ⊂ S * and A 0 coincides with the kernel of the restriction of Γ 0 . The Weyl function corresponding to this restricted boundary triplet is a compression of the original Weyl function onto a subspace of G. In the following proposition this is made precise from the point of view of an orthogonal decomposition of G.
Proposition 2.5.12. Let S be a closed symmetric relation in H and assume that {G, Γ 0 , Γ 1 } is a boundary triplet for S * with A 0 = ker Γ 0 , and corresponding γfield γ and Weyl function M . Assume that G has the orthogonal decomposition with corresponding orthogonal projections P and P and canonical embedding ι . Then the following statements hold: is closed and symmetric with S ⊂ S ⊂ A 0 .
(ii) The adjoint (S ) * of S is given by is a boundary triplet for (S ) * such that A 0 = ker Γ 0 .
(iv) The γ-field γ and Weyl function M corresponding to the boundary triplet {G , Γ 0 , Γ 1 } are given by Moreover, for every closed symmetric extension S with S ⊂ S ⊂ A 0 there exists an orthogonal decomposition (2.5.21) of G such that (2.5.22) holds.

Proof. (i) & (ii) It is clear from the definition that S ⊂ S ⊂ A 0 and that S can be written as
Hence, S = A Θ when Θ = {0} × G . It follows that S is closed, and by (1.3.4) one has Θ * = G × G, so that Theorem 2.1.3 (iv) shows (iii) With the choice f, g ∈ (S ) * one has Γ 0 f = P Γ 0 f and Γ 0 g = P Γ 0 g. Then (2.1.1) yields (2.5.23) It follows from the surjectivity of Γ and the identity Γ 0 f = P Γ 0 f when f ∈ (S ) * , that Together with (2.5.23) this shows that {G , Γ 0 , Γ 1 } is a boundary triplet for (S ) * . It is clear that A 0 = ker Γ 0 holds.
Finally, if S is a closed symmetric extension of S with the property S ⊂ A 0 , then S = A Θ for some closed symmetric relation Θ in G such that Θ ⊂ {0} × G by Theorem 2.1.3 (v). As Θ is closed, there exists a closed subspace G ⊂ G such that Θ = {0} × G . With G = (G ) ⊥ it is clear that the orthogonal decomposition (2.5.21) of G holds and S is of the form (2.5.22).
In the situation of Proposition 2.5.12 the intermediate extensions of S can also be interpreted as intermediate extensions of S. In the next corollary the connection between these extensions relative to the appropriate boundary triplets is explained.
Corollary 2.5.13. Assume that the parameter space G has the orthogonal decomposition (2.5.21)and let S be as in Proposition 2.5.12. Let Θ be a closed relation in G and let Θ be the closed linear relation in G defined by (2.5.24)

For the intermediate extensions induced by Θ and Θ one has
Proof. It is clear that the relation Θ defined in (2.5.24) is a closed relation in G.
The identity in (2.5.25) now follows from where (2.5.24) has been used in conjunction with the boundary triplet in Proposition 2.5.12.
Let S and S be closed symmetric relations in H and H which are unitarily equivalent, and let {G, Γ 0 , Γ 1 } and {G , Γ 0 , Γ 1 } be boundary triplets for S * and (S ) * , respectively. The notion of unitary equivalence for these boundary triplets will now be introduced which leads to unitary equivalence of the corresponding extensions, γ-fields and Weyl functions.
Let H and H be Hilbert spaces and let U ∈ B(H, H ) be a unitary operator from H onto H . Let S and S be closed symmetric relations in H and H , respectively, such that they are unitarily equivalent by means of U , that is, (2.5.26) in the sense of Definition 1.3.7. It follows from (1.3.7) that this assumption is equivalent to S * and (S ) * being equivalent under U , Then U maps N λ (S * ) unitarily onto N λ ((S ) * ), and hence Furthermore, let V ∈ B(G, G ) be a unitary mapping from G onto G . Then the closed relations Θ in G and Θ in G are unitarily equivalent if The notion of unitary equivalence of two boundary triplets involves not only the unitary equivalence between H and H , but also the unitary equivalence between G and G .
In the next proposition it will be shown that for unitarily equivalent boundary triplets the corresponding closed extensions, γ-fields, and Weyl functions are unitarily equivalent.  U ∈ B(H, H ) and  V ∈ B(G, G ). Then the following statements hold: (i) For all closed relations Θ in G and Θ in G connected via (2.5.27) the closed extensions are unitarily equivalent by means of U ∈ B(H, H ), that is, (ii) The γ-fields γ and γ corresponding to

Proof. (i) It follows from the definition that
Since S and S are unitarily equivalent, so are S * and (S ) * , and hence one has {g, g } ∈ (S ) * if and only if {g, g } = {Uf, Uf } for some {f, f } ∈ S * . Thus, by the unitary equivalence of the boundary triplets one obtains (ii) By item (i), the self-adjoint relations A 0 = ker Γ 0 and A 0 = ker Γ 0 are unitarily equivalent by means of U , which implies that ρ(A 0 ) = ρ(A 0 ), and hence the γ-fields γ and γ are defined on the same subset of C. For λ ∈ ρ(A 0 ) one computes (iii) Since ρ(A 0 ) = ρ(A 0 ), the Weyl functions M and M are defined on the same subset of C. Fix λ ∈ ρ(A 0 ), let ψ ∈ G and choose {f λ , λf λ } ∈ N λ (S * ) such that Γ 0 {Uf λ , λUf λ } = ψ. Since {Uf λ , λUf λ } ∈ N λ ((S ) * ), it follows from the definition of the Weyl function and (2.5.28) that Let S and S be closed symmetric relations in H and H with boundary triplets {G, Γ 0 , Γ 1 } and {G , Γ 0 , Γ 1 } that are unitarily equivalent by means of a unitary operator U ∈ B(H, H ) and a unitary operator V ∈ B(G, G ) as in Proposition 2.5.15. Then according to Proposition 2.5.15 one has that cf. (2.5.28). In particular, this implies that the multivalued parts of A 0 and A 0 are connected by mul A 0 = U (mul A 0 ), and since U is unitary it also follows that The following corollary is an immediate consequence of Proposition 2.5.15 (ii) and Proposition 2.3.2 (ii).
In Theorem 4.2.6 it will be shown that if S and S are simple (see Section 3.4) and their Weyl functions are unitarily equivalent, then in fact the corresponding boundary triplets are unitarily equivalent.

Kreȋn's formula for intermediate extensions
Let S be a closed symmetric relation in a Hilbert space H and assume that {G, Γ 0 , Γ 1 } is a boundary triplet for S * . According to Theorem 2.1.3, the mapping Γ = (Γ 0 , Γ 1 ) induces a bijective correspondence between the set of (closed) intermediate extensions A Θ of S and the set of (closed) relations Θ in G, via and A 0 = ker Γ 0 corresponds to Θ = {0}×G. For λ ∈ ρ(A 0 ) the relation (A Θ −λ) −1 will be regarded as a perturbation of the resolvent of the self-adjoint extension A 0 of S. This fact is expressed by the formula provided in Theorem 2.6.1 and some variants under the additional assumption λ ∈ ρ(A Θ ) are discussed afterwards. In the special case λ ∈ ρ(A Θ ) one has (A Θ −λ) −1 ∈ B(H) and the resolvent difference, and hence also the perturbation term, are bounded operators. Moreover, it is shown later how the different types of spectral points λ ∈ σ(A Θ ) which are contained in ρ(A 0 ) are related to the Weyl function and the parameter Θ. A more in-depth treatment of the connection of the spectrum and the Weyl function can be found in Chapter 3.
In the next theorem the difference of (A Θ − λ) −1 and (A 0 − λ) −1 , λ ∈ ρ(A 0 ), is expressed in a perturbation term which involves the Weyl function M and the parameter Θ. This results in a general version of Kreȋn's formula for intermediate extensions.
Theorem 2.6.1. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , A 0 = ker Γ 0 , and let γ and M be the corresponding γ-field and Weyl function, respectively. Moreover, let Θ be a closed relation in G and let be the corresponding extension via (2.1.5). Then for all λ ∈ ρ(A 0 ) one has the equality where the inverses in the first and the last term are taken in the sense of relations.
and (2.6.2) holds in the sense of bounded linear operators.
Proof. Assume that λ ∈ ρ(A 0 ). In order to establish the identity (2.6.2) it must be shown that the relations on the left-hand side and right-hand side coincide. First the inclusion (⊂) in (2.6.2) will be shown. For this purpose, consider and hence g Θ − g 0 ∈ N λ (S * ). Since γ(λ) maps G onto N λ (S * ) there exists an element ϕ ∈ G such that g Θ = g 0 + γ(λ)ϕ.
Assume that λ ∈ ρ(A 0 ). In Theorem 2.6.1 it is shown that then λ ∈ ρ(A Θ ) leads to (Θ − M (λ)) −1 ∈ B(G). In fact, there is a one-to-one correspondence between the part of the spectrum of A Θ contained in ρ(A 0 ) and the spectrum of Θ − M (λ) contained in ρ(A 0 ). The following theorem and its corollary are direct consequences of the Kreȋn formula (2.6.2). A complete description of the spectrum of self-adjoint extensions A Θ in terms of the singularities of the function λ → (Θ − M (λ)) −1 is given in Section 3.8.
Theorem 2.6.2. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , A 0 = ker Γ 0 , and let γ and M be the corresponding γ-field and Weyl function, respectively. Moreover, let Θ be a closed relation in G and let be the corresponding extension via (2.1.5). Then the following statements hold for all λ ∈ ρ(A 0 ): , and in this case (2.6.7) Proof. Assume that λ ∈ ρ(A 0 ) and consider the right-hand side of (2.6.2) as the sum of the operator (A 0 − λ) −1 ∈ B(H) and the relation Hence, the domain of the right-hand side of (2.6.2) is given by where it was used that γ(λ) ∈ B(G, H). Thus, it follows from (2.6.2) that (2.6.8) Due to the definition of the sum of relations and mul (A 0 − λ) −1 = {0} the multivalued part of the right-hand side of (2.6.2) is given by Thus, it follows from (2.6.2) that (2.6.9) The proof of the theorem is based on the identities (2.6.8) and (2.6.9). For the interpretation of (2.6.8) recall that γ(λ), λ ∈ ρ(A 0 ), maps G isomorphically onto N λ (S * ); see Proposition 2.3.2 (i). This implies that the restriction γ(λ) * : N λ (S * ) → G is an isomorphism.
The Kreȋn formula (2.6.2) was formulated above in terms of the closed relation Θ in the Hilbert space G. Now the form of the Kreȋn formula will be given when a tight parametric representation of Θ is chosen; cf. Section 1.10. where E is a Hilbert space and A, B ∈ B(E, G), and assume that this representation of Θ is tight, i.e., ker A ∩ ker B = {0} holds. Then for all λ ∈ ρ(A 0 ) one has and in this case (2.6.13) Proof. According to Theorem 2.6.2, for λ ∈ ρ(A 0 ) one has that is, λ ∈ ρ(A Θ ) if and only if (Θ − M (λ)) −1 ∈ B(G). Due to the tightness of the representation (2.6.12), Lemma 1.11.6 shows that as for all λ ∈ ρ(A 0 ) one has that M (λ) ∈ B(G). In this case it follows that (Θ − M (λ)) −1 = A(B − M (λ)A) −1 . Furthermore, the resolvent formula (2.6.13) follows from (2.6.2).
Let again Θ be a closed relation in G and assume, in the same way as at the end of Section 2.2, that Θ admits an orthogonal decomposition (2.6.14) into a (not necessarily densely defined) operator part Θ op acting in the Hilbert space G op = dom Θ * = (mul Θ) ⊥ and a multivalued part Θ mul = {0} × mul Θ in the Hilbert space G mul = mul Θ; cf. Section 1.3. Recall that, in particular, closed symmetric, self-adjoint, (maximal) dissipative, and (maximal) accumulative relations Θ in G admit such a decomposition.
Corollary 2.6.4. Assume that the closed relation Θ in Theorem 2.6.1 has the orthogonal decomposition (2.6.14), let P op be the orthogonal projection onto G op , and denote the canonical embedding of G op into G by ι op . Let Proof. In view of (2.6.14), one sees that for This shows that Θ op − P op M (λ)ι op is a bijective closed operator in G op . Hence, with respect to the decomposition (2.6.14). Now the identity is an immediate consequence. This together with Theorem 2.6.1 implies the statement.
If the closed relation Θ in G admits a decomposition of the form as in Corollary 2.5.13, where Θ is a closed relation in the Hilbert space G and G = G ⊕G , then Kreȋn's formula can also be interpreted in the context of the intermediate symmetric extension S of S in Proposition 2.5.12 and the corresponding restriction of the boundary triplet {G, Γ 0 , Γ 1 }. More precisely, if Θ is of the form (2.6.15) and S and the boundary triplet {G , Γ 0 , Γ 1 } are as in Proposition 2.5.12 with corresponding γ-field γ and Weyl function M , and then A Θ = ker (Γ 1 − ΘΓ 0 ) = A Θ and A 0 = ker Γ 0 = ker Γ 0 = A 0 hold by Corollary 2.5.13 and Proposition 2.5.12, respectively. Moreover, by Theorem 2.6.1 one has (Θ − M (λ)) −1 ∈ B(G ) for all λ ∈ ρ(A Θ ) ∩ ρ(A 0 ) and In the special case where Θ = Θ op and {0} × G = Θ mul as in (2.6.14) one has so that Kreȋn's formula in Corollary 2.6.4 can be rewritten in the form The behavior of Kreȋn's formula under transformations of boundary triplets will be discussed next. To this end suppose that S is a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , A 0 = ker Γ 0 , and let γ and M be the corresponding γ-field and Weyl function, respectively. Consider a closed extension according to Theorem 2.6.1. Let G be a further Hilbert space and assume that W ∈ B(G × G, G × G ) satisfies the identities in (2.5.1). Let {G , Γ 0 , Γ 1 } be the corresponding transformed boundary triplet in (2.5.2) with γ-field γ and Weyl function M specified in Proposition 2.5.3. Let A 0 = ker Γ 0 and define the closed relation Θ in G by Θ = W[Θ]; cf. (2.5.4). By Proposition 2.5.2, one has and hence for all λ ∈ ρ(A Θ ) ∩ ρ(A 0 ) Kreȋn's formula in Theorem 2.6.1 has the form In this sense Kreȋn's formula is invariant under transformations of boundary triplets.
Next, Theorem 2.6.2 will be complemented for the case where the extensions are self-adjoint. Recall from Section 1.5 that for a self-adjoint relation H a spectral point λ ∈ R belongs to the discrete spectrum σ d (H) if λ is an eigenvalue with finite multiplicity which is an isolated point in σ(H). It will be used that λ ∈ σ d (H) if and only if dim ker (H − λ) < ∞ and ran (H − λ) = ran (H − λ). (2.6.16) The complement of the discrete spectrum of H in σ(H) is the essential spectrum, denoted by σ ess (H).
Theorem 2.6.5. Let S be a closed symmetric relation, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , A 0 = ker Γ 0 , and let M be the corresponding Weyl function. Let Θ be a self-adjoint relation in G and let be the corresponding self-adjoint extension via (2.1.5). Then the following statements hold for all λ ∈ ρ(A 0 ): Proof. Here one relies on the observations made in the proof of Theorem 2.6.2. Assume that λ ∈ ρ(A 0 ).

Kreȋn's formula for exit space extensions
The Kreȋn formula in Theorem 2.6.1 holds for intermediate extensions of a symmetric relation S in a Hilbert space H. In particular, these intermediate extensions contain maximal dissipative, maximal accumulative, and self-adjoint extensions. Now consider larger Hilbert spaces K which contain H as a closed subspace and self-adjoint relations A in K which extend S as studied by Kreȋn and Naȋmark. It will be shown that such self-adjoint extensions induce families of relations in H which also extend S. For these families of relations there is a version of the Kreȋn formula, which will also be called Kreȋn-Naȋmark formula in this text.
In the present context theŠtraus family and the compressed resolvent appear when one considers self-adjoint extensions of a closed symmetric relation S in a Hilbert spaces H. Let the Hilbert space H ⊕ H be an extension of H, where the Hilbert space H is an exit space. Assume that the self-adjoint relation A in H ⊕ H is an extension of the symmetric relation S in H, i.e., S ⊂ A. TheŠtraus family and the compressed resolvent of A consist of relations in the closed subspace H that extend S in the following sense.
The Kreȋn-Naȋmark formula in the following theorem is now an immediate consequence.
Theorem 2.7.4. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , A 0 = ker Γ 0 , and let γ and M be the corresponding γfield and Weyl function, respectively. Let A be a self-adjoint extension of S in H⊕H . Then with the Nevanlinna family τ in G from Theorem 2.7.3 the compressed resolvent R(λ) in (2.7.2) of A is given by the Kreȋn-Naȋmark formula (2.7.9) Proof. As in the proof of Theorem 2.7.3, it follows from Theorem 2.6.1 that for λ ∈ C \ R one has Hence, the formula (2.7.9) follows from (2.7.4).
In Chapter 4 the converse of Theorem 2.7.4 will be proved: for every Nevanlinna family in G there exists a self-adjoint exit space extension A of S such that (2.7.9) holds for the compressed resolvent of A.
Just as in the case of Corollary 2.6.3, there is now a formulation of the Kreȋn formula for exit space extensions in terms of a parametric representation of the Nevanlinna family τ . Assume that the Nevanlinna pair {A, B} is a tight representation of the Nevanlinna family τ ; cf. Section 1.12. Then the next corollary can be shown in the same way as Corollary 2.6.3 by applying Proposition 1.12.6.
TheŠtraus family in Theorem 2.7.3 can also be described in terms of a representing Nevanlinna pair {A, B}.
Corollary 2.7.6. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , and let T (λ), λ ∈ C \ R, be theŠtraus family in Theorem 2.7.3. Let the corresponding Nevanlinna family τ have the tight representation τ = {A, B} with the Nevanlinna pair {A, B}. Then (2.7.10) Proof. By assumption τ (λ), λ ∈ C \ R, is given as In the following a particular self-adjoint exit space extension of S will be studied. Here the exit space is the parameter space G.
Proposition 2.7.7. Let S be a closed symmetric relation in H and let {G, Γ 0 , Γ 1 } be a boundary triplet for S * . Then is a self-adjoint extension of S in H ⊕ G. The correspondingŠtraus family T (λ), λ ∈ C \ R, in H has the form T (λ) = f ∈ S * : −Γ 1 f = λΓ 0 f = ker Γ 1 + λΓ 0 (2.7.12) and the compressed resolvent R(λ) onto H is given by a self-adjoint relation A in H ⊕ H will be viewed as a self-adjoint extension of the trivial symmetric relation S in H . TheŠtraus family T (λ), λ ∈ C, in H corresponding to A in H ⊕ H is defined by which can be viewed as the Kreȋn-Naȋmark formula in H for the extension A of S . For the self-adjoint relation A in Proposition 2.7.7 it turns out in this new context that the correspondingŠtraus family in G is given by the function −M .
Proposition 2.7.8. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * , and let M be the corresponding Weyl function. Consider the self-adjoint relation Then the correspondingŠtraus family in G is given by and coincides with −M (λ), λ ∈ C \ R. Furthermore, the compressed resolvent of A to G is given by (2.7.18) here P G : H ⊕ G → G denotes the orthogonal projection from H ⊕ G onto G and ι G : G → H ⊕ G is the canonical embedding of G into H ⊕ G.
Proof. It follows from the definition of theŠtraus family in (2.7.1) that theŠtraus family corresponding to A in the Hilbert space G has the form (2.7.17).
Since {Γ 0 f, −Γ 1 f } belongs to (2.7.17) if and only if f ∈ N λ (S * ), it is also clear that for all λ ∈ C \ R theŠtraus family coincides with the values −M (λ) of the Weyl function corresponding to the boundary triplet {G, Γ 0 , Γ 1 }. The formula (2.7.18) follows from (2.7.16) in this special case.

Perturbation problems
Let A be a self-adjoint relation in the Hilbert space H, let V ∈ B(H) be a bounded self-adjoint operator in H, and consider the self-adjoint relation this follows from Lemma 1.11.2 with H = A, R = λ − V and S = λ. In particular, if V in (2.8.1) belongs to some left-sided or right-sided operator ideal, then the same is true for the difference of the resolvents of A and B. From this point of view perturbation problems in the resolvent sense are more general than additive perturbations of the form (2.8.1). Such perturbation problems embed naturally in the framework of the extension theory that has been discussed in this chapter.
In the next theorem the particularly simple case of finite-rank perturbations is treated. where (2.8.2) was used in the last equality. Now Theorem 1.7.8 implies that A and B are transversal self-adjoint extensions of S. Theorem 2.5.9 shows that there exists a boundary triplet {C n , Γ 0 , Γ 1 } such that (2.8.3) holds, and the formula (2.8.4) follows from Theorem 2.6.1. One also concludes from (2.8.4) and the fact that M (λ) is bijective for λ ∈ ρ(A) ∩ ρ(B) (see Corollary 2.5.4) that the difference of the resolvents in (2.8.2) is of rank n for all λ ∈ ρ(A) ∩ ρ(B). The last statement on the eigenvalues of B follows from Theorem 2.6.2 (i).
The following result is a generalization of Theorem 2.8.1 that applies to nonself-adjoint intermediate extensions B. Proof. It is clear that S = A ∩ B is a closed symmetric relation and hence there exists a boundary triplet {G, Γ 0 , Γ 1 } for S * such that A = ker Γ 0 ; cf. Theorem 2.4.1.
Since B is a closed extension of S, there exists a closed relation Θ in G such that B = ker (Γ 1 − ΘΓ 0 ). By construction, the relations A and B are disjoint and hence it follows from Proposition 2.1.8 (i) that Θ is a closed operator in G. The resolvent formula and the assertion on the spectrum of B are immediate consequences of Theorem 2.6.1 and Theorem 2.6.2.
Proof. Assume that (2.8.10) holds and that ρ(Θ) = ∅. As Θ 0 = {0} × G is the self-adjoint relation in G which corresponds to A = ker Γ 0 and (Θ 0 − ξ) −1 = 0 for ξ ∈ C, one concludes from Theorem 2.8.3 and (2.8.10) that Together with the assumption that (2.8.10) is of infinite rank, this implies that Θ is an unbounded closed operator in G.
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