Operator Models for Nevanlinna Functions

The classes of Weyl functions and more generally of Nevanlinna functions will be studied from the point of view of reproducing kernel Hilbert spaces.

Note that the first two items in this definition are not independent. Nonnegativity is the stronger condition.
The kernels described in Definition 4.1.1 form the basis of the theory of reproducing kernel Hilbert spaces. They arise naturally in the following context. Let Ω ⊂ C be an open set and let (H, ·, · ) be a Hilbert space of functions defined on Ω with values in a Hilbert space G. The Hilbert space H is called a reproducing kernel Hilbert space if for all μ ∈ Ω the operation of point evaluation is bounded. In other words, for each μ ∈ Ω the linear operator E(μ) : H → G, defined by E(μ)f = f (μ), belongs to B(H, G).
In the next theorem a kernel is related to a Hilbert space of functions in which point evaluation is bounded.
(v) If the G-valued functions in H are holomorphic on Ω, then K(·, ·) is holomorphic and uniformly bounded on compact subsets of Ω.
(iii) To see that K(·, ·) is a nonnegative kernel it suffices to observe that the matrix is nonnegative. Lemma 4.1.2 implies that K(·, ·) is symmetric.
(v) To see that K(·, ·) is uniformly bounded on compact subsets of Ω, note first that K(λ, λ) = E(λ)E(λ) * = E(λ) 2 . (4.1.3) Now observe that for all f ∈ H and ϕ ∈ G, Since by assumption the function λ → f (λ) from Ω to G is holomorphic, it follows that the mapping λ → E(λ) from Ω to B(H, G) is holomorphic, which implies that λ → E(λ) is continuous. Hence, for any compact set K ⊂ Ω there is some Therefore, (4.1.3) shows that the kernel K(·, ·) is uniformly bounded on compact subsets of Ω.
In particular, one has by (4.1.5) for all f ∈H(K) Thus, the definition of the form ·, · in (4.1.6) implies the reproducing kernel property in (4.1.7); the kernel K(·, ·) is called a reproducing kernel, relative to the linear spaceH(K) in (4.1.4). It will now be shown that the form defined by (4.1.5) or (4.1.6) is actually a scalar product.
Lemma 4.1.4. Let Ω ⊂ C be an open set, let G be a Hilbert space, and let the kernel K(·, ·) in (4.1.1) be nonnegative. Define the spaceH(K) by (4.1.4) and define the form ·, · onH(K) as in (4.1.5) and (4.1.6). ThenH(K) is a pre-Hilbert space with the scalar product ·, · .
The assumption that the kernel K(·, ·) is nonnegative means that the n × n matrix is nonnegative. Thus for a typical element f = n j=1 α j K(·, ν j )ϕ j ∈H (K) one sees that f, f ≥ 0 and hence ·, · is a nonnegative symmetric sesquilinear form onH(K). In particular, ·, · satisfies the Cauchy-Schwarz inequality. This implies that ·, · is positive definite. In fact, if f, f = 0 for some f ∈H(K), then | f, g | 2 ≤ f, f g, g = 0 for all g ∈H(K).
In the following theorem it is shown that a nonnegative kernel K(·, ·) on Ω produces a Hilbert space H(K), as a completion ofH(K), of functions on Ω for which point evaluation is a continuous map. Moreover, if the kernel is holomorphic and uniformly bounded on compact subsets of Ω, then the functions in the resulting Hilbert space are holomorphic.
Theorem 4.1.5. Let G be a Hilbert space, let K(·, ·) be a nonnegative kernel on the open set Ω ⊂ C, and let the form ·, · onH(K) be defined as in (4.1.5) and (4.1.6). Then the following statements hold: (i) The completion H(K) of the pre-Hilbert space (H(K), ·, · ) can be identified with a Hilbert space of G-valued functions defined on Ω.
(ii) For f ∈ H(K) one has the reproducing kernel property is a continuous linear mapping and (iv) If the kernel K(·, ·) is holomorphic and uniformly bounded on every compact subset of Ω, then the functions in H(K) are holomorphic on Ω.
Proof. (i) Let (H(K), ·, · ) be the Hilbert space that is obtained when one completes the pre-Hilbert space (H(K), ·, · ). It will be shown that the elements in H(K) can be identified with G-valued functions on Ω. For this let f ∈ H(K) and fix some λ ∈ Ω. Consider the functional Then an application of the Cauchy-Schwarz inequality shows that and hence Ψ f,λ is continuous. By the Riesz representation theorem, there is a unique vector ψ f,λ ∈ G such that Let F(Ω, G) be the space of all G-valued functions defined on Ω, and consider the mapping It follows from the definition of ι and ψ f,λ that (4.1.10) and this equality also shows that ι is a linear mapping. The mapping ι in (4.1.9) is injective. To see this, assume that ι(f ) = 0 for some f ∈ H(K). This means ι(f )(λ) = 0 for all λ ∈ Ω, and (4.1.10) implies f, K(·, λ)ϕ = 0 for all λ ∈ Ω and ϕ ∈ G. Since the linear span of the functions K(·, λ)ϕ forms the dense subspaceH(K) of H(K), it follows that f = 0, that is, ι is injective.
Observe that for f ∈H(K) it follows from (4.1.10) and the reproducing kernel property (4.1.7) that for all λ ∈ Ω, and hence ι(f ) = f for f ∈H(K). In other words, ι restricted to the dense subspaceH(K) is the identity, so that ι(H(K)) =H(K). Finally, item (i) follows when the subspace ran ι of F(Ω, G) is equipped with the scalar product induced by H(K), that is, forf,g ∈ ran ι define Then ι is a unitary mapping from the Hilbert space (H(K), ·, · ) onto the Hilbert space (ran ι, ·, · ∼ ).
(ii) After identifying ι(f ) and f ∈ H(K) as in (i), the reproducing kernel property is immediate from (4.1.10).
(iii) With the identification from (ii) observe that for all λ ∈ Ω and ϕ ∈ G the mapping f → (f (λ), ϕ) G (4.1.11) is continuous on H(K). In fact, this follows from (ii) and the computation For a fixed λ ∈ Ω the mapping is closed. To see this, suppose that f n → f in H(K) and E(λ)f n → ψ in G. As dom E(λ) = H(K), it follows that f ∈ dom E(λ) and the continuity of (4.1.11) then yields This shows E(λ)f = ψ and hence E(λ) is a closed operator. Since dom E(λ) = H(K), the closed graph theorem implies that E(λ) is continuous. It remains to check the identity K(λ, μ) = E(λ)E(μ) * for λ, μ ∈ Ω. For this let ϕ, ψ ∈ G, λ, μ ∈ Ω, and note that E(μ) * ϕ ∈ H(K) is a function in the variable λ. Hence, E(λ)E(μ) * ϕ = (E(μ) * ϕ)(λ), the reproducing kernel property, and the symmetry of the kernel K(·, ·) imply By assumption, the functions K(·, μ)ϕ are holomorphic on Ω, and hence so are the functions f n . Now let K ⊂ Ω be a compact set and let sup λ∈K K(λ, λ) = M K . Then for λ ∈ K one gets and hence (f n (·), ϕ) G → (f (·), ϕ) G uniformly on K for all ϕ ∈ G. As K is an arbitrary compact subset of Ω, it follows that the function λ → (f (λ), ϕ) G is holomorphic on Ω for all ϕ ∈ G. This implies that f is holomorphic.
If the kernel K(·, ·) is holomorphic and uniformly bounded on every compact subset of Ω, then the elements in the reproducing kernel Hilbert space H(K) can be described as holomorphic functions from Ω to G which satisfy an additional boundedness condition involving the kernel K(·, ·).
Theorem 4.1.6. Let G be a Hilbert space, assume that K(·, ·) is a nonnegative holomorphic kernel on the open set Ω ⊂ C which is uniformly bounded on every compact subset of Ω, and let (H(K), ·, · ) be the associated reproducing kernel Hilbert space. Then f ∈ H(K) with f ≤ γ if and only if f : Ω → G is holomorphic and the n × n matrix is nonnegative for all n ∈ N, ν 1 , . . . , ν n ∈ Ω, and ϕ 1 , . . . , ϕ n ∈ G.
Assume that the function f : Ω → G is holomorphic and there exists γ > 0 such that the n×n matrix (4.1.12) is nonnegative for all n ∈ N, ν 1 , . . . , ν n ∈ Ω, and ϕ 1 , . . . , ϕ n ∈ G. Together with (4.1.13) and (4.1.14), this implies that the relation from H(K) to C, spanned by the elements where ν 1 , . . . , ν n ∈ Ω, ϕ 1 , . . . , ϕ n ∈ G, and α 1 , . . . , α n ∈ C, is a bounded functional with bound γ. Furthermore, it is densely defined on H(K), so that it admits a uniquely defined bounded linear extension defined on all of H(K). This functional is represented by a unique element F ∈ H(K) with F ≤ γ via the Riesz representation theorem. In particular this means that whereas by the reproducing kernel property one has Combining the last two identities one concludes that f = F , which gives that f ∈ H(K) and f ≤ γ.
Due to the holomorphy it is sometimes convenient to consider a set of functions λ → K(λ, μ)ϕ, ϕ ∈ G, on a determining set of points μ ∈ Ω.
Corollary 4.1.7. Let K(·, ·) be a nonnegative holomorphic kernel on an open set Ω ⊂ C which is uniformly bounded on every compact subset of Ω, and let H(K) be the associated reproducing kernel Hilbert space. Let D ⊂ Ω be a set of points which has an accumulation point in each connected component of Ω. Then Proof. The inclusion (⊃) is obvious from (4.1.4). To show the inclusion (⊂), it suffices to verify that the linear space . Therefore, let f ∈ H(K) be orthogonal to this set. Then 0 = f, K(·, μ)ϕ = (f (μ), ϕ) G for all μ ∈ D and ϕ ∈ G, and hence f (μ) = 0 for all μ ∈ D. Since f ∈ H(K) is holomorphic on Ω, the assumption on D now implies that f (λ) = 0 for all λ ∈ Ω. Hence, f = 0 and the proof is complete.
Let K(·, ·) be a nonnegative holomorphic kernel on an open set Ω. If Ω ⊂ C is an open set such that Ω ⊂ Ω and if K (·, ·) is a nonnegative holomorphic kernel on Ω extending K(·, ·), then the functions in the reproducing kernel Hilbert space H(K) may be seen as restrictions to Ω of the functions in the reproducing kernel Hilbert space H(K ).
Proposition 4.1.8. Let K(·, ·) be a nonnegative holomorphic kernel on an open set Ω ⊂ C which is uniformly bounded on every compact subset of Ω. Assume that Ω ⊂ C is an open set such that Ω ⊂ Ω and that K (·, ·) is a nonnegative holomorphic kernel on Ω which is uniformly bounded on every compact subset of Ω and which is equal to K(·, ·) on Ω. Then Proof. Consider the linear space of functions from Ω into G generated by K (·, ·) viaH It is clear that the analogous linear space is contained inH(K ) in the sense that each function λ → K(λ, μ)ϕ with μ ∈ Ω is the restriction to Ω of the function λ → K (λ, μ)ϕ. Hence, a continuity argument shows the inclusion For the opposite inclusion consider f | Ω : Ω → C for some f ∈ H(K ), and set γ = f . Then, by Theorem 4.1.6, the matrix is nonnegative for all n ∈ N, ν 1 , . . . , ν n ∈ Ω , and ϕ 1 , . . . , ϕ n ∈ G. In particular, is nonnegative for all n ∈ N, ν 1 , . . . , ν n ∈ Ω, and ϕ 1 , . . . , ϕ n ∈ G. Another application of Theorem 4.1.6 implies f | Ω ∈ H(K).
Under suitable circumstances multiplication of a given reproducing kernel by an operator function gives rise to a new reproducing kernel. In the following proposition this fact and the relation between the corresponding reproducing kernel Hilbert spaces are explained.
Proposition 4.1.9. Let G be a Hilbert space, assume that K(·, ·) is a nonnegative kernel on Ω, and let (H(K), ·, · ) be the associated reproducing kernel Hilbert space. Let Φ : Ω → B(G) be such that 0 ∈ ρ(Φ(λ)) for all λ ∈ Ω. Then is a nonnegative kernel on Ω and the corresponding reproducing Hilbert space (H(K Φ ), ·, · Φ ) is unitarily equivalent to H(K) via the mapping Moreover, if K(·, ·) is holomorphic and uniformly bounded on every compact subset of Ω, and Φ is holomorphic, then also K Φ (·, ·) is holomorphic and uniformly bounded on every compact subset of Ω.
Proof. The definition (4.1.15) leads to the identity Hence, the nonnegativity of K(·, ·) implies that K Φ (·, ·) is a nonnegative kernel. Moreover, (4.1.15) shows that for all μ ∈ Ω and ϕ ∈ G which is valid for all μ, ν ∈ Ω and all ϕ, ψ ∈ G, shows that the mapping M Φ fromH(K) ontoH(K Φ ) is an isometry. Its unique bounded linear extension gives a unitary mapping from H(K) onto H(K Φ ). In order to see that this extension M Φ acts as multiplication by Φ on all functions in H(K), let f ∈ H(K) and choose a sequence f n ∈H(K) such that f n → f in H(K). By isometry, the sequence Hence, taking limits one sees that and therefore The last assertion on the holomorphy and uniform boundedness of K Φ (·, ·) on compact subsets of Ω is clear.

Realization of uniformly strict Nevanlinna functions
The aim of this section is to show that every operator-valued uniformly strict Nevanlinna function can be realized as the Weyl function corresponding to a boundary triplet. The reproducing kernel Hilbert space associated with a given uniformly strict Nevanlinna function will serve as a model space. The uniqueness of the model is discussed as well.
Let G be a Hilbert space and let M be a B(G)-valued Nevanlinna function. The associated Nevanlinna kernel (4.2.1) and N M (λ, λ) = M (λ), λ ∈ C \ R. Then clearly the kernel N M is symmetric. The kernel N M is holomorphic, since is holomorphic for each μ ∈ C \ R. Moreover, from the definition of N M one sees immediately that In the next theorem it turns out that the kernel N M is, in fact, nonnegative on C \ R. Note also that the kernel N M is uniformly bounded on compact subsets of C \ R since Proof. The function M has the integral representation with self-adjoint operators α, β ∈ B(G), β ≥ 0, and a nondecreasing self-adjoint operator function Σ : where the integral in (4.2.2) converges in the strong topology; cf. Theorem A.4.2. For any n ∈ N, points λ 1 , . . . , λ n ∈ C \ R, and elements ϕ 1 , . . . , ϕ n ∈ G it follows from (4.2.2) that . (4.2. 3) The first matrix on the right-hand side in (4.2.3) is nonnegative as for any vector (x 1 , . . . , x n ) ∈ C n and ϕ = x i ϕ i the nonnegativity of the operator β implies To see that the second matrix on the right-hand side in (4.2.3) is also nonnegative, first use Proposition A.3.7 and Proposition A.3.4 to obtain and when max |t l − t l−1 | tends to zero these finite Riemann-Stieltjes sums converge to Hence, also (4.2.4) is nonnegative; thus, both matrices on the right-hand side in (4.2.3) are nonnegative, and so is their sum, i.e., the kernel N M is nonnegative.
According to Theorem 4.1.5, the nonnegative kernel N M gives rise to a Hilbert space of holomorphic G-valued functions, which will be denoted by H(N M ), with inner product ·, · ; cf. Section 4.1. Recall that the reproducing kernel property holds for all functions f ∈ H(N M ). The main results in this section concern a Nevanlinna function M and the construction of a self-adjoint relation which represents M in a sense to be explained. The construction will involve the associated reproducing kernel space H(N M ).
Let M be a (not necessarily uniformly strict) B(G)-valued Nevanlinna function. The first main objective in this section is the contruction of a minimal model in which the function is realized as the compressed resolvent of a self-adjoint relation. The uniqueness of the construction will be discussed after the theorem. Note that the definition of the self-adjoint relation involves multiplication by the independent variable; however, the resulting functions do not necessarily belong to H(N M ). Then is a self-adjoint relation in the Hilbert space H(N M ) ⊕ G and the compressed resolvent of A onto G is given by Furthermore, the self-adjoint relation A satisfies the following minimality condition: Step 1. The relation A in (4.2.6) contains an essentially self-adjoint relation.
It follows from the definition of N M in (4.2.1) that Therefore, one sees from (4.2.6) that B ⊂ A. It remains to show that B is essentially self-adjoint. The symmetry of B is easily verified: it follows from the definition in (4.2.1) and the reproducing kernel property (4.2.5) that Therefore, also the elements of the form , see Corollary 4.1.7, it follows from (4.2.9) and (4.2.10) that Now let B be the closure of the symmetric relation B. It is clear that B is symmetric and that ran (B −λ 0 ) is closed (see Proposition 1.4.4 and Lemma 1.2.2). Hence, it follows from the above considerations that ran (B − λ 0 ) = H(N M ) ⊕ G, and Theorem 1.5.5 yields that B is self-adjoint in H(N M ) ⊕ G.
Step 2. The relation A is self-adjoint. To prove this, it suffices to establish that the closure of B coincides with the relation A.
First one shows that the relation A is closed. To see this, let where the sequence on the left-hand side belongs to A, so that Then taking limits in the last identity and using the continuity of point evaluation in H(N M ), see Theorem 4.1.5, leads to the identity Then f, f ∈ H(N M ), ϕ, ϕ ∈ G, and For an element in B of the form Step 3. It remains to establish the identities (4.2.7) and (4.2.8). Both are direct consequences of (4.2.6). In fact, let λ ∈ C \ R and note that by Theorem 4.1.5 and Corollary 4.1.7. This implies (4.2.8).
The model and the self-adjoint relation in Theorem 4.2.2 are unique up to unitary equivalence. This is a consequence of the following general equivalence result.
Theorem 4.2.3. Let G, H, and H be Hilbert spaces and let A and A be self-adjoint relations in the product spaces H ⊕ G and H ⊕ G, respectively. Denote by P G and P G the orthogonal projections from H ⊕ G and H ⊕ G onto G, respectively, and let ι G and ι G be the corresponding canonical embeddings. Assume that A satisfies the minimality condition and that A satisfies the minimality condition Furthermore, assume that (4.2.14) Then A and A are unitarily equivalent, that is, there exists a unitary operator Proof. Note that the elements of the form . . , n, and n ∈ N are arbitrarily chosen, form a dense subspace of the Hilbert space H ⊕ G by the assumption (4.2.12). Likewise, the elements of the form . . , n , and n ∈ N are arbitrarily chosen, form a dense subspace of the Hilbert space H ⊕ G by the assumption (4.2.13). Define the linear relation U from H ⊕ G to H ⊕ G as the linear span of all pairs of the form where ϕ j , ψ j ∈ G, α j , β j ∈ C, λ j ∈ C \ R for j = 1, . . . , n, and n ∈ N are arbitrarily chosen. Then according to (4.2.15) and (4.2.16) the relation U has a dense domain and a dense range. To show that the relation U is isometric, i.e., h = h for all {h, h } ∈ H, one has to verify that To see this, it suffices to observe that (4.2.14) implies and, likewise, by symmetry, Moreover, using the resolvent identity, one sees that for λ j = λ i (4.2.14) implies and a limit argument together with the continuity of the resolvent shows that the same is true in the case λ j = λ i . Thus, the relation U is isometric; hence it is a well-defined isometric operator and the closure of U , denoted again by U , is a unitary operator from H ⊕ G onto H ⊕ G.
Next it will be shown that A and A are unitarily equivalent under U . To see this, one needs to show and by the continuity of the resolvent the same relation holds also for λ = λ j . Thus, (4.2.17) has been established.
The second main result in this section concerns a minimal model for uniformly strict Nevanlinna functions. By means of Theorem 4.2.2 it will be shown that every uniformly strict Nevanlinna function M is the Weyl function corresponding to a boundary triplet of a simple symmetric operator in the Hilbert space H(N M ). After this result it will be shown that every boundary triplet producing the same Weyl function is unitarily equivalent to the boundary triplet in this construction. Note that the description of (S M ) * involves functions which do not necessarily belong to H(N M ); however the definition of S M concerns functions which remain in H(N M ) after multiplication by the independent variable.
is a closed simple symmetric operator in H(N M ) and its adjoint is given by Moreover, the mappings and the corresponding Weyl function is given by M .
Proof. By Theorem 4.2.2, the relation Step 1. The relation S M in (4.2.18) is a closed symmetric operator with adjoint (S M ) * given by (4.2.19). First observe that the relation S M in (4.2.18) satisfies Observe that Hence, to conclude (4.2.19), it suffices to show that T M is closed. To see this, By Theorem 4.1.5 (iii), the point evaluation is continuous, so that Hence, and using that Im M (λ 0 ) is boundedly invertible (since M is assumed to be uniformly strict), it follows that ϕ n → ϕ and hence also ϕ n → ϕ for some ϕ, ϕ ∈ G. Therefore, In other words, {f, f } ∈ T M , and hence T M is closed.
Step 2. The mappings in (4.2.20) form a boundary triplet for (S M ) * . First note that they are single-valued. Indeed, assume that {f, f } ∈ (S M ) * is the trivial element, then M (ξ)ϕ − ϕ = 0 for the corresponding elements ϕ, ϕ ∈ G and all ξ ∈ C \ R. Taking ξ = λ 0 and ξ = λ 0 with λ 0 ∈ C \ R and using the fact that ker (Im M (λ 0 )) = {0}, one concludes that ϕ = 0 and ϕ = 0. To verify the abstract Green identity, Since A is self-adjoint and thus symmetric, one sees that Thus, the abstract Green identity is satisfied. It remains to show that Γ maps onto G 2 , which then implies that {G, Γ 0 , Γ 1 } is a boundary triplet for (S M ) * . First, it will be shown that ran Γ is dense in G 2 . Suppose that {α , α} ∈ G 2 is orthogonal to ran Γ, that is, and hence M (ξ)α + α = 0 for all ξ ∈ C \ R. Now as above one concludes that α = 0 and α = 0. Therefore, ran Γ is dense in G 2 . Next, it will be shown that ran Γ is closed. For this consider again the selfadjoint relation A as a subspace of (H(N M ) ⊕ G) 2 and define the orthogonal projections P and I − P by respectively. Then P A = T M = (S M ) * is closed and hence, by Lemma C.4, is closed. Since A is self-adjoint, it follows from this and (1.3.5) that and hence (4.2.23) is closed. In other words, ran (Γ 0 , −Γ 1 ) is closed or, equivalently, ran (Γ 0 , Γ 1 ) is closed.
Step 3. The γ-field and Weyl function corresponding to the boundary triplet (4.2.20) are given by γ in (4.2.21) and M , respectively, and the symmetric operator S M in (4.2.18) is simple.
As this is true for all f λ ∈ N λ ((S M ) * ) and all λ ∈ C \ R, one concludes that M is the Weyl function corresponding to the boundary triplet (4.2.20).
To compute the γ-field and to show that S M is simple, assume again that f λ ∈ N λ ((S M ) * ). Then (4.2.24) with ξ = λ implies that Hence, the γ-field corresponding to the boundary triplet (4.2.20) is given by (4.2.21), and since the elements  The construction of the boundary triplet in Theorem 4.2.4 is unique up to unitary equivalence. More precisely, if S is a simple symmetric operator in a Hilbert space H and there is a boundary triplet for S * with the same Weyl function M as in Theorem 4.2.4, then the boundary triplets are unitarily equivalent in the sense of Definition 2.5.14 (where G = G ). This is a consequence of the following general equivalence result, which is a further specification of Theorem 4.2.3.
Proof. The basic idea of the proof follows the proof of Theorem 4.2.3. By assumption, the Weyl functions M and M of the boundary triplets {G, Γ 0 , Γ 1 } and {G, Γ 0 , Γ 1 } coincide. It follows from Proposition 2.3.6 (iii) that the corresponding γ-fields γ and γ satisfy the identity for all λ, μ ∈ C \ R, λ = μ, and γ(λ) * γ(λ) = γ (λ) * γ (λ) follows by continuity. Define the linear relation U from H to H as the linear set of all pairs of the form where ϕ j ∈ G, α j ∈ C, λ j ∈ C \ R for j = 1, . . . , n, and n ∈ N are arbitrarily chosen. It is clear from the definition of U that its domain is given by and its range is given by  (4.2.27) it follows that for each λ ∈ C \ R the restriction U : ker (S * − λ) → ker ((S ) * − λ) is unitary. Thus, (4.2.27) implies by Proposition 2.3.6 (ii) that and, in particular, for all λ ∈ C \ R and ϕ ∈ G.
Now let A 0 = ker Γ 0 and A 0 = ker Γ 0 . Then the property (4.2.27) and Proposition 2.3.2 (ii) imply for all λ, μ ∈ C \ R, and hence Since S is simple, one sees that span {ran γ(μ) : μ ∈ C \ R} is a dense subspace of H and hence follow. Therefore, by Lemma 1.3.8, the self-adjoint relations A 0 and A 0 are unitarily equivalent, that is, This immediately yields for some g ∈ H , so that by means of (4.2.29), Proposition 2.3.2 (iv), and (4.2.27) one obtains that To see that the boundary triplets are unitarily equivalent, first recall that A 0 and A 0 are unitarily equivalent, see (4.2.30), and that cf. (4.2.27). The direct sum decompositions (S ) * = A 0 + N λ ((S ) * ) and S * = A 0 + N λ (S * ) (4.2.33) for λ ∈ ρ(A 0 ) = ρ(A 0 ) from Theorem 1.7.1 now show that Together with (4.2.34) this shows that the boundary triplets are unitarily equivalent. Let According to Theorem 2.5.1 and Proposition 2.5.3 every operator matrix with the properties (2.5.1) gives rise to a boundary triplet {G, is a closed simple symmetric operator in H(N M ) and its adjoint is given by (4.2.40) The corresponding boundary triplet {G, (Γ M ) 0 , (Γ M ) 1 } is given by Proof. To establish (4.2.42), note that and hence, in view of (4.2.38) and (4.2.43), Therefore, according to Proposition 4.1.9, each {F, where (4.2.47) was used in the last equality.
Moreover, from (4.2.35) and the model in Theorem 4.2.4 one then concludes or, equivalently, This representation is in fact the counterpart of the observations concerning Weyl functions in Proposition 2.7.8.

Realization of scalar Nevanlinna functions via L 2 -space models
In the case of a scalar Nevanlinna function M one may also construct a minimal model via the corresponding integral representation (4.3.1) Here the constants α, β, and the nondecreasing function σ satisfy It is a consequence of this representation that Therefore, a scalar Nevanlinna function M is equal to the real constant α if and only if M is not uniformly strict, i.e., if and only if Im M (λ) = 0 for some, and hence for all λ ∈ C \ R. Under the assumption that the Nevanlinna function is not constant, a model involving the integral representation is constructed in this section. Moreover, a concrete natural isomorphism between the new model space and the reproducing kernel Hilbert space H(N M ) in Theorem 4.2.4 will be given.
The new model is build in the Hilbert space L 2 dσ (R) consisting of all (equivalence classes of) complex dσ-measurable functions f such that R |f | 2 dσ < ∞, equipped with the scalar product The following observations will be used in the construction of the model. Under the integrability condition on σ in (4.3.2) one has that t 1 + t 2 , and, in addition, It is first assumed for convenience that the linear term in the integral representation (4.3.1) is absent, that is, β = 0. The general case β = 0 will be discussed afterwards in Theorem 4.3.4. The usual notation for general elements {f, f } will also be used here; the reader should be aware that f is not the derivative here.
is a closed simple symmetric operator in L 2 dσ (R) and its adjoint is given by Moreover, the mappings are well defined and {C, Γ 0 , Γ 1 } is a boundary triplet for S * . The corresponding γ-field is given by the mapping

3.8)
and the corresponding Weyl function is M .
Proof. The proof consists of two steps. In Step 1 it will be shown that S in (4.3.5) is a closed symmetric operator and that {C, Γ 0 , Γ 1 } is a boundary triplet for its adjoint S * in (4.3.6). Moreover, it will be shown that the γ-field is given by (4.3.8) and that M is the corresponding Weyl function. In Step 2 the simplicity of S is concluded from Corollary A.1.5.
Step 1. The right-hand side of (4.3.6) is a relation which satisfies the conditions in Theorem 2.1.9. To see this, denote the relation on the right-hand side of (4.3.6) by T and think of Γ 0 and Γ 1 as being defined on T . Observe that the mapping Γ 1 is well defined thanks to (4.3.3). Likewise, the operator S is well defined due to (4.3.4). First, it is clear that A 0 = ker Γ 0 ⊂ T is the maximal multiplication operator by the independent variable in L 2 dσ (R): Since A 0 is a self-adjoint operator in L 2 dσ (R), one sees that condition (i) in Theorem 2.1.9 is satisfied.
Next, one has that Γ is surjective. For this, note that, by (4.3.6), the defect subspace N λ (T ), λ ∈ C \ R, of T consists of elements of the form which after a simple computation gives or, in other words, Using (4.3.10) one now observes that This shows that Γ is surjective; just note that λ ∈ C \ R implies that Im M (λ) = 0. Hence, condition (ii) in Theorem 2.1.9 is satisfied. Finally, the abstract Green identity for T and Γ in Theorem 2.1.9 (iii) will be exhibited. For this purpose, let f = {f, f }, g = {g, g } ∈ T , and assume that tf (t) − f (t) = c and tg(t) − g (t) = d for some c, d ∈ C. Then a calculation shows that The last line gives, after substitution of c and d, . Hence, also condition (iii) in Theorem 2.1.9 is satisfied. Therefore, all conditions of Theorem 2.1.9 have been verified. As a consequence the relation ker Γ 0 ∩ ker Γ 1 is closed and symmetric. It coincides with S in (4.3.5), since for f ∈ T one has Thus, it follows from Theorem 2.1.9 that the adjoint of the closed symmetric operator S in (4.3.5) is given by T and hence has the form (4.3.6), and, moreover, {C, Γ 0 , Γ 1 } is a boundary triplet for S * . As a byproduct of (4.3.9) and (4.3.10) one sees that the corresponding γ-field is given by (4.3.8) and that the corresponding Weyl function coincides with M .
Step 2. It remains to show that the operator S in (4.3.5) is simple. To see this, assume that there is an element g ∈ L 2 dσ (R) which is orthogonal to all elements f λ ∈ ker (S * − λ), λ ∈ C \ R, that is, Then g = 0 in L 2 dσ (R) by Corollary A.1.5. Thus, the linear span of the defect spaces ker (S * −λ), λ ∈ C \ R, is dense in L 2 dσ (R) and now Corollary 3.4.5 implies that the symmetric operator S is simple. This completes the proof of Theorem 4.3.1.
Note that in the model in Theorem 4.3.1 the self-adjoint extension A 0 is equal to the operator of multiplication by the independent variable. The closed minimal operator S is not densely defined if and only if the constant functions belong to L 2 dσ (R) or, equivalently, σ is a finite measure. According to Theorem 4.2.6 the L 2 dσ (R)-space model for the function M and the model in Theorem 4.2.4 are unitarily equivalent thanks to the simplicity of the underlying symmetric operators. A concrete unitary map will be provided in the following proposition.
The γ-field corresponding to the boundary triplet in Theorem 4.3.1 at a point λ ∈ C \ R is given by the mapping It follows from the integral representation (4.3.1) with β = 0 that In view of (4.3.11) and (4.3.13), it follows that The special case of a rational Nevanlinna function serves as an illustration of Theorem 4.3.1. In this situation the measure dσ in (4.3.1) has only finitely many point masses and the space L 2 dσ (R) can be identified with C n . Example 4.3.3. Let α 1 ∈ R, n ∈ N, γ 1 , . . . , γ n > 0, and −∞ < δ 1 < δ 2 < · · · < δ n < ∞, and consider the rational complex Nevanlinna function (4.3.14) Define a nondecreasing step function σ : R → [0, ∞) by and consider the corresponding L 2 -space L 2 dσ (R) with the scalar product The rational Nevanlinna function N in (4.3.14) admits the integral representation as in (4.3.1), where via the unitary mapping and the maximal operator of multiplication by the independent variable in L 2 dσ (R) is unitarily equivalent to the diagonal matrix the simple symmetric operator S in Theorem 4.3.1 is unitarily equivalent to the restriction of the diagonal matrix in (4.3.15) to the orthogonal complement of the subspace span (1, . . . , 1) . Furthermore, S * corresponds to the relation According to Theorem 4.3.1, the corresponding Weyl function is the rational Nevanlinna function N in (4.3.14).
Now the general case of a scalar Nevanlinna function of the form (4.3.1) with β > 0 will be addressed.

Theorem 4.3.4. Let M be a scalar Nevanlinna function of the form (4.3.1) with β > 0. Then
is a closed simple symmetric operator in L 2 dσ (R) ⊕ C and its adjoint is given by Moreover, for f ∈ S * the mappings are well defined and {C, Γ 0 , Γ 1 } is a boundary triplet for S * . The corresponding γ-field is given by the mapping and the corresponding Weyl function is given by M .
Proof. The proof is similar to the one of Theorem 4.3.1, thus only a brief sketch will be given. Denote the right-hand side of the formula for S * by T and think of Γ 0 and Γ 1 as being defined on T . It is clear that A 0 = ker Γ 0 ⊂ T is the orthogonal componentwise sum of the maximal multiplication operator by the independent variable in L 2 dσ (R) and the purely multivalued part {0} × C. Hence, A 0 is a selfadjoint relation. To show that Γ is surjective, note that the defect subspace N λ (T ), λ ∈ C \ R, of T consists of elements of the form Again, as in the proof of Theorem 4.3.1 it follows that Γ is surjective. It can be checked by straightforward calculation as in the proof of Theorem 4.3.1 that the abstract Green identity is satisfied. Thus, by Theorem 2.1.9, one concludes that T is the adjoint of the closed symmetric relation S and that Γ 0 and Γ 1 define a boundary triplet for S * . Hence, the statements about the γ-field and the Weyl function follow.
To show the simplicity of S, assume that there is an element orthogonal to N λ (S * ) for all λ ∈ C \ R, i.e., there exists an element g ∈ L 2 dσ (R) and a constant γ ∈ C such that For λ = iy and y → ∞ it follows that γ = 0 and hence Corollary A.1.5 implies g = 0. Therefore, the closed linear span of all N λ (S * ), λ ∈ C \ R, is equal to L 2 dσ (R) ⊕ C, and, as a consequence, the closed symmetric operator S is simple.
In the situation of Theorem 4.3.4 one sees that S is a nondensely defined operator and that mul S * is spanned by the vector Now A 0 is the only self-adjoint extension of S which is multivalued: it is the orthogonal sum of multiplication by the independent variable in L 2 dσ and the space spanned by (4.3.17).  Proof. The proof is similar to the one of Proposition 4.3.2 and will be sketched briefly. Recall that the γ-field corresponding to the boundary triplet in Theorem 4.2.4 at a point λ ∈ C \ R is given by the mapping 3.18) while the γ-field corresponding to the boundary triplet in Theorem 4.3.4 at a point λ ∈ C \ R is given by the mapping It follows from the integral representation (4.3.1) that Hence, (4.3.18)-(4.3.19) and the fact that imply that the property (4.2.25) holds. This implies that the boundary triplets in Theorem 4.3.4 and Theorem 4.2.4 are unitarily equivalent.
In the following it is briefly explained how the self-adjoint multiplication operator in L 2 dσ (R) and the model discussed in this section (in the case β = 0) are connected with the spectral theory and the limit properties of the Weyl function in Chapter 3. For this assume that σ : R → R is a nondecreasing function such that R 1 1 + t 2 dσ(t) < ∞ and consider the self-adjoint multiplication operator in L 2 dσ (R). Then it is known from Example 3.3.7 that the spectrum σ(A 0 ) coincides with the set of growth points of the function σ, see (3.2.1), and the same is true for the absolutely continuous part σ ac , singular continuous part σ sc , and singular part σ s of σ. On the other hand, the one-dimensional restriction of A 0 in Theorem 4.3.1 is a closed simple symmetric operator in L 2 dσ (R) and {C, Γ 0 , Γ 1 } in (4.3.7) is a boundary triplet for S * in (4.3.6) with A 0 = ker Γ 0 and corresponding Weyl function (4.3.20) where α is an arbitrary real number in the definition of the boundary map Γ 1 in (4.3.7). Hence, the results on the description of the spectrum of A 0 via the limit properties of the Weyl function from Section 3.5 and Section 3.6 apply in the present situation. For example, Theorem 3.6.5 shows that which is also clear from Theorem 3.2.6 (i), taking into account (3.1.25) and Corollary 3.1.8 (ii). Similar observations can be made for the other spectral subsets.
In other words, in the special situation where A 0 is the self-adjoint multiplication operator in L 2 dσ (R) the general description of the spectrum of A 0 and its subsets in Chapter 3 in terms of the limit properties of the associated Weyl function in (4.3.20) agrees with Example 3.3.7.

Realization of Nevanlinna pairs and generalized resolvents
In this section the model from Section 4.2 for Nevanlinna functions will be extended to the general setting of Nevanlinna pairs and of generalized resolvents. As a byproduct the extended model leads to the Sz.-Nagy dilation theorem.
Let G be a Hilbert space and let {A, B} be a Nevanlinna pair of B(G)-valued functions; cf. Section 1.12. The associated Nevanlinna kernel N A,B (4.4.1) and is holomorphic for each μ ∈ C \ R, that is, the kernel N A,B is holomorphic. Moreover, it follows from (4.4.1) and Definition 1.12.3 that In the next theorem it is shown that the kernel N A,B is, in fact, nonnegative on C \ R. Note also that the kernel N A,B is uniformly bounded on compact subsets of C \ R since Proof. To see this, let N be a uniformly strict B(G)-valued Nevanlinna function and let ε > 0. Then εN is again a uniformly strict Nevanlinna function. Define the function S ε by By Proposition 1.12.6, S ε is a Nevanlinna function. A calculation shows that the Nevanlinna kernel associated with the function S ε is of the form Observe that for any ε > 0 the kernel N Sε is nonnegative since S ε is a Nevanlinna function. The identity (4.4.2) shows that the kernel is nonnegative for any ε > 0.
To show that the kernel N A,B is nonnegative, assume the contrary, i.e., assume that N A,B is not nonnegative. Then it follows from the definition of nonnegativity that there exist n ∈ N, λ 1 , . . . , λ n ∈ C \ R, elements ϕ 1 , . . . , ϕ n ∈ G, and a vector c ∈ C n , such that Since −x > 0 and the kernel N N is nonnegative, one can choose ε > 0 so small Combining these results one arrives at the inequality which contradicts the nonnegativity of the kernel in (4.4.3). Thus, the kernel N A,B is nonnegative.
Let {A, B} be a Nevanlinna pair in G. According to Theorem 4.1.5, with the nonnegative kernel N A,B there is associated a Hilbert space of holomorphic G-valued functions, which will be denoted by H (N A,B ), with inner product ·, · ; cf. Section 4.1. Recall that the reproducing kernel property is a self-adjoint relation in the Hilbert space H (N A,B ) ⊕ G and the compressed resolvent of A A,B onto G is given by Furthermore, the self-adjoint relation A A,B satisfies the following minimality condition: (4.4.5) Proof. The proof is almost the same as the proof of Theorem 4.2.2; therefore, only the main elements are recalled and the details are left to the reader.
Step 1. Use the Nevanlinna pair {A, B} to define the auxiliary relation B in It is a direct computation to show that B ⊂ A A,B . Likewise, by a similar computation one verifies that B is symmetric in Therefore, choosing μ = λ 0 and taking into account that ran (B(λ 0 ) + λ 0 A(λ 0 )) = G by Definition 1.12.3 and Lemma 1.12.5 it follows that {0} ⊕ G ⊂ ran (B − λ 0 ); hence also the elements of the form belong to ran (B − λ 0 ). It follows from Corollary 4.1.7 that ran (B − λ 0 ) is dense in H(N A,B ) ⊕ G, and thus B is essentially self-adjoint.

Since the compressed resolvent of A
The following special case is of interest. and for λ, μ ∈ C \ R the corresponding Nevanlinna kernel can be written as Let again {A, B} be a Nevanlinna pair, let τ be the corresponding Nevanlinna family, and consider a Nevanlinna pair {C, D} which is equivalent to {A, B} via (4.4.9), so that it generates the same Nevanlinna family τ . Then according to Theorem 4.4.2, is a self-adjoint relation in the Hilbert space H(N C,D ) ⊕ G and the compressed resolvent of A C,D onto G is given by Furthermore, the self-adjoint relation A C,D satisfies the following minimality condition: The explicit connection between the various models involving these kernels in  Proof. In the identity (4.4.10) set Φ(λ) = X(λ) * with X(λ) as in (4.4.9). Since X(λ) is boundedly invertible one may apply Proposition 4.1.9 and hence U in (4.4.12) is unitary. Now consider an element This implies One verifies in the same way that every element can be written in the form This shows that the self-adjoint relations A A,B and A C,D are unitarily equivalent under the mapping U ; cf. Definition 1.3.7.
The discussions in this section so far centered mainly on Nevanlinna pairs and will now be put in a slightly different context. Definition 4.4.5. Let H be a Hilbert space and let R be a B(H)-valued function defined on C \ R. Then R is a called a generalized resolvent if it has the following properties: With the function R one associates the kernel R R R R (·, ·) : Ω × Ω → B(H), (4.4.13) and R(λ, λ) = R (λ) − R(λ) 2 , λ ∈ C \ R. Then clearly the kernel R R is symmetric. Since λ → R(λ) is holomorphic, the mapping λ → R R (λ, μ) is holomorphic for each μ ∈ C \ R, that is, the kernel R R is holomorphic. Note also that the kernel R R is uniformly bounded on compact subsets of C \ R since For R R to be a reproducing kernel in the sense of Theorem 4.1.5 one needs nonnegativity. Since R is a generalized resolvent, a straightforward computation shows that {C, D} is a Nevanlinna pair and that the kernels satisfy (4.4.14) cf. (4.4.1) and (4.4.13). Now it follows from Theorem 4.4.1 (with G = H) that the kernel R R (·, ·) is nonnegative.
Let R : C \ R → B(H) be a generalized resolvent. By Theorem 4.1.5, the corresponding nonnegative kernel R R induces a Hilbert space of holomorphic Hvalued functions, which will be denoted by H(R R ), with inner product ·, · ; cf. Section 4.1. Recall that the reproducing kernel property holds for all functions f ∈ H(R R ). The following result gives a representation of the function R.
is a self-adjoint relation in the Hilbert space H(R R ) ⊕ H and the compressed resolvent of A R onto H is given by Furthermore, the self-adjoint relation A R satisfies the following minimality condition: is a self-adjoint relation in the Hilbert space H(R R ) ⊕ H and that its compressed resolvent is given by where the fact that τ (λ)+ λ = {−R(λ), I} was used in the last equality. Moreover, the minimality condition (4.4.15) holds.
By Corollary 4.4.7, every generalized resolvent can be interpreted as a compressed resolvent of a self-adjoint relation. Such compressed resolvents have been discussed briefly in the context of the Kreȋn formula in Section 2.7 and will be further studied in Section 4.5. The next theorem complements Corollary 4.4.7 by providing equivalent conditions. In particular, generalized resolvents or, equivalently, compressed resolvents, are characterized as Stieltjes transforms of nondecreasing families of nonnegative contractions. As a simple consequence one obtains the Sz.-Nagy dilation theorem in Corollary 4.4.9. (i) The function R is a generalized resolvent.

(ii) There exist a Hilbert space K and a self-adjoint relation
Furthermore, the self-adjoint relation A satisfies the following minimality condition: (iii) There exists a nondecreasing function Σ : R → B(H), whose values are nonnegative contractions, such that R dΣ(t) ∈ B(H), R dΣ(t) ≤ 1, and Proof. (i) ⇒ (ii) This follows directly from Corollary 4.4.7.
(ii) ⇒ (iii) Since A is self-adjoint, one can write with the spectral measure E(·) of A; cf. (1.5.6). The function t → E((−∞, t)) is a nondecreasing family of orthogonal projections from R to B(H ⊕ K) and one has that Now define Σ(t) = P H E((−∞, t))ι H , which is a nondecreasing family of nonnegative contractions from R to B(H) that satisfies R dΣ(t) ∈ B(H) and the estimate R dΣ(t) ≤ 1. (iii) ⇒ (i) It is clear that the function R : C \ R → B(H) is holomorphic and satisfies R(λ) = R(λ) * for λ ∈ C \ R. Moreover, it follows from Proposition A.5.4 that which implies that R is a generalized resolvent.
The next corollary is a variant of the dilation theorem, which goes back to M.A. Naȋmark and B. Sz.-Nagy; here it is obtained from Theorem 4.4.8 and the Stieltjes inversion formula. Then there exist a Hilbert space K and a left-continuous nondecreasing function E : R → B(H ⊕ K), whose values are orthogonal projections, such that Proof. Associate with Σ the function (4.4.16) By Theorem 4.4.8, there exists a Hilbert space K and a self-adjoint relation A in H ⊕ K such that the compression of the resolvent of A onto H is given by (4.4.17) Let E(·) be the spectral measure of A and let t → E((−∞, t)) be the corresponding spectral function, which is left-continuous and satisfies lim t→−∞ E((−∞, t)) = 0. As in the proof of Theorem 4.4.8 one has Taking into account (4.4.16) and (4.4.17), it follows that and hence for all h ∈ H holds. This is Kreȋn's formula for the compressed resolvents of self-adjoint exit space extensions (as studied by M. A. Naȋmark); it is also referred to as Kreȋn-Naȋmark formula in this text; cf. Section 2.7. The goal of this section is to show the converse statement. More precisely, it will be proved that for every Nevanlinna family τ (λ), λ ∈ C \ R, in the Hilbert space G there exists a self-adjoint exit space extension A of S such that the compressed resolvent of A onto H is given by the Kreȋn-Naȋmark formula. The following result is a first step.
Lemma 4.5.1. Let S be a closed symmetric relation in H, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * with A 0 = ker Γ 0 , and let γ and M be the corresponding γ-field and Weyl function, respectively. Let τ = {A, B} be a Nevanlinna family in G and define R(λ), λ ∈ C \ R, by Then the kernel (4.5.4) In particular, the kernel R R is nonnegative, symmetric, holomorphic, and uniformly bounded on compact subsets of C \ R. Proof.
Lemma 4.5.1 shows that R R (·, ·) is a reproducing kernel. Therefore, one may apply Theorem 4.4.8.

Orthogonal coupling of boundary triplets
In this section a different look is taken at the Kreȋn-Naȋmark formula. By means of an abstract coupling method for direct orthogonal sums of symmetric relations and corresponding boundary triplets, a particular self-adjoint extension A of the direct sum is identified, and it is shown that the compressed resolvent of A is of the same form as in the Kreȋn-Naȋmark formula. When combined with Theorem 4.2.4, this coupling procedure provides a constructive approach to the exit space extension in Theorem 4.5.2 in the special case where the Nevanlinna family τ is a uniformly strict Nevanlinna function.
First a slightly more general, abstract point of view is adopted. In the following let S and T be closed symmetric relations in the Hilbert spaces H and H , respectively, and assume that the defect numbers of S and T coincide: n + (S) = n − (S) = n + (T ) = n − (T ) ≤ ∞.
The compressions of the resolvent of the self-adjoint relation A in (4.6.4) to H and H are of interest. Note that the resolvent of A 0 in (4.6.2) is given by the direct orthogonal sum of the resolvents of A 0 and B 0 , and hence for λ ∈ ρ( A 0 ) the compressions to H and H are respectively. The next statement follows directly from Proposition 4.6.1 and (4.6.3).
Corollary 4.6.2. Let S and T be closed symmetric relations in the Hilbert spaces H and H with boundary triplets {G, Γ 0 , Γ 1 } and {G, Γ 0 , Γ 1 }, and corresponding γ-fields and Weyl functions γ, γ and M, τ , respectively. Then for all λ ∈ C \ R the following statements hold: (i) The compression of the resolvent of the self-adjoint relation A in (4.6.4) to H is given by (ii) The compression of the resolvent of the self-adjoint relation A in (4.6.4) to H is given by Corollary 4.6.2 and Proposition 4.6.1 can also be viewed as an alternative approach to the Kreȋn-Naȋmark formula in the special case where the Nevanlinna family τ in Theorem 4.5.2 is a uniformly strict Nevanlinna function. In fact, according to Theorem 4.2.4 every uniformly strict B(G)-valued Nevanlinna function can be realized as a Weyl function, that is, there exist a (reproducing kernel) Hilbert space H (= H(N τ )), a closed simple symmetric operator T (= S τ ) in H , and a boundary triplet {G, Γ 0 , Γ 1 } for the adjoint T * such that τ is the corresponding Weyl function. In this situation the relation A in (4.6.4) is self-adjoint in H ⊕ H = H ⊕ H(N τ ) and its compressed resolvent in Corollary 4.6.2 coincides with the one in the Kreȋn-Naȋmark formula in Theorem 4.5.2. Summing up, the following special case of Theorem 4.5.2 is a consequence of the coupling method in Proposition 4.6.1 and Corollary 4.6.2.
Corollary 4.6.3. Let S be a closed symmetric relation, let {G, Γ 0 , Γ 1 } be a boundary triplet for S * with A 0 = ker Γ 0 , and let γ and M be the corresponding γ-field and Weyl function, respectively. Let τ be a uniformly strict B(G)-valued Nevanlinna function. Then there exist an exit Hilbert space H and a self-adjoint relation A in H ⊕ H such that A is an extension of S and the compressed resolvent of A is given by the Kreȋn-Naȋmark formula:
With (4.6.9) one then concludes that which in turn yields (4.6.8).
In the next proposition a particular boundary triplet {G ⊕ G, Γ 0 , Γ 1 } is specified such that the self-adjoint relation A in (4.6.4) coincides with the kernel of the boundary mapping Γ 0 . The corresponding Weyl function M is useful for the spectral analysis of A; cf. Chapter 6.