Extensions of symmetric operators I: The inner characteristic function case

Given a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation $B$ with equal indices and inner Livsic characteristic function $\Theta _B$ by constructing a natural bijection between the set of self-adjoint extensions and the set of all contractive analytic functions $\Phi$ which are greater or equal to $\Theta _B$. In addition we characterize the set of all symmetric extensions $B'$ of $B$ which have equal indices in the case where $\Theta _B$ is inner.


Introduction
The purpose of this paper is to study of the family of all closed symmetric extensions of a given closed simple symmetric linear transformation B with equal deficiency indices (n, n), 1 ≤ n < ∞ defined on a domain in a separable Hilbert space in the case where the Livsic characteristic function of B is an inner function. For n ∈ N ∪ {∞}, S n (H) will denote the set of all closed simple symmetric linear transformations with indices (n, n) defined in a separable Hilbert space H. More generally S n will denote the family of all closed simple symmetric linear transformations with indices (n, n) defined in some separable Hilbert space, and S the set of all closed simple symmetric linear transformations with equal indices defined in some separable Hilbert space.
If A is a symmetric linear transformation which extends B ∈ S n (H) and Dom(A) is also contained in H then we call A a canonical extension of B. If, on the other hand A is symmetric in K where K H, then we call A a non-canonical extension of B. The set of all canonical extensions of B can be completely characterized by the set of all partial isometries between the deficiency subspaces Ker(B * − i) and Ker(B * + i), see for example [1,Chapter VII]. Our goal is to provide a new characterization the set of extensions, canonical and non-canonical in the special case where the characteristic function Θ B is inner. (Recall that in this case B is unitarily equivalent to multiplication by z in a model subspace K 2 ΘB = H 2 (C + ) ⊖ Θ B H 2 (C + ) of Hardy space [2,3,4].) We will begin with the study of the self-adjoint extensions of B, denoted Ext(B), and show that there is a bijective correspondence between A ∈ Ext(B) and the set of all contractive analytic (matrix) functions Φ A which obey: see Theorem 8.14. Here, given contractive analytic matrix functions Φ, Θ on C + , we say that Θ ≤ Φ provided that Θ −1 Φ is contractive and analytic on C + . This provides an alternative to the classical results of M.G. Krein (see e.g [5, Theorem 6.5] for the (1, 1) case) which are formulated in terms of generalized resolvents and R-functions. Our characterization has the advantage of providing a natural function-theoretic connection between the Livsic characteristic function of B ∈ S and the set of its self-adjoint extensions.
We will also study a natural partial order on S : we say that B 1 where ≃ denotes unitary equivalence and we use the ⊂ notation to denote when one linear transformation is an extension of another. In words, B 1 is less than or equal to B 2 if B 2 is an extension of B ′ 1 where B ′ 1 is unitarily equivalent to B 1 . Application of the Cayley transform, which is a bijection from S onto V , the set of all partial isometries with equal indices, converts this into a partial order on V . Modulo unitary equivalence, this is the same as the partial order previously defined by Halmos and McLaughlin on partial isometries in [6]. In the case where Θ B1 is an inner function, we provide necessary and sufficient conditions on Θ B2 so that B 1 B 2 in Theorem 9.5.
Many of these results will be achieved using the concept of a generalized model. This is a reproducing kernel Hilbert space theory approach which generalizes the concept of a model for a symmetric operator as defined in [4].

Preliminaries
Recall that a linear transformation B is simple, symmetric and closed with deficiency indices (n, n) if it is defined on a domain Dom(B) contained in a separable Hilbert space H and has the following properties: Bx, y = x, By , ∀x, y ∈ Dom(B), B is symmetric; Condition (2.2) can be restated equivalently as: B is simple if and only if there is no non-trivial subspace reducing for B such that the restriction of B to the intersection of its domain with this subspace is self-adjoint. For many of our results we will need to assume that n < ∞ is finite.
A partial isometry V is called simple, or c.n.u. (completely non-unitary) if it has no unitary restriction to a proper (and non-trivial) reducing subspace. The deficiency indices for V are the pair of non-negative integers (n + , n − ) defined by n + := dim (Ker(V )) and n − := dim Ran (V ) ⊥ , and it is not difficult to see that these are the same as the defect indices of V as defined in [7].
There is a bijective correspondence between S n (H) and V n (H) which we now describe: Given a simple symmetric linear transformation B ∈ S n (H) and z ∈ C \ R, let Q z denote the projection onto Ran (B − z). The Cayley transform V B of B is the partial isometry Conversely suppose that V is a simple partial isometry on H with defect indices (n + , n − ). One can construct a symmetric linear transformation B V by defining and Again it is easy to check that B V and V have the same deficiency indices. One can further verify that B VB = B and V BV = V for any symmetric linear transformation B and partial isometry V , respectively. This shows that the maps B → V B and V → B V are inverses of each other so that these maps are bijections between S and V . We will use this bijection between V and S to formulate problems in whichever setting is most convenient, and to obtain equivalent results for both classes of linear transformations.
Given The Livsic characteristic function is then [8] Θ B (z) := b(z)B −1 (z)A(z), (2.8) and this can be shown to be a contractive n × n matrix-valued analytic function on C + , the upper half-plane. Note that the characteristic function Θ B always vanishes at z = i. Different choices of bases in the definition yield a new characteristic function Θ B which is related to the first by where R, Q are fixed unitary matrices. Two Livsic characteristic functions Θ 1 , Θ 2 are said to coincide or to be equivalent if they are related in this way.
For most of this paper we will assume that Θ B is an inner function, i.e. Θ B has non-tangential boundary values on R almost everywhere with respect to Lebesgue measure, and these non-tangential boundary values are unitary matrix-valued. In this case B ≃ Z ΘB , where Z ΘB ∈ S K 2 ΘB is the symmetric operator of multiplication by z on the domain Dom(Z ΘB ) = {f ∈ K 2 ΘB | zf ∈ K 2 ΘB }, in the model space K 2 ΘB = H 2 ⊖ Θ B H 2 , and here H 2 = H 2 (C + ) is the Hardy space of the upper half-plane.
As shown in [4], one can also define the Livsic characteristic function for the case where n = ∞, and this new definition coincides with the old one for n < ∞. As first shown by M.S. Livsic, the Livsic characteristic function is a complete unitary invariant for S n (H): It will also be convenient to define Ext U (B) to be the set of all self-adjoint linear transformations A on K for which A ∈ Ext(U BU * ) for some isometry U : H → K.
The set Ext(B) is called the set of extensions of B. In the case where K = H, we say that A is a canonical self-adjoint extension of B. Recall that the canonical self-adjoint extensions A of B can all be obtained by first computing the Cayley transform V := b(B), extending this by a rank − n isometry U : Dom(V ) ⊥ → Ran (V ) ⊥ to obtain a unitary extension V U of V , and then taking the inverse Cayley transform to obtain a self-adjoint linear transformation A := b −1 (V U ).

Linear relations
In the case where B is not densely defined, its adjoint B * is not a linear operator. Instead B * can be realized as a linear relation, and we will dicuss the basic facts about linear relations that will be needed in this section. The material from this section is taken primarily from [10] and [11,Section 1.1]. A linear relation L is defined to be a subspace of H ⊕ H. Note that L = G(T ) is the graph of some closed linear operator T provided that L is closed and (0, f ) ∈ L implies that f = 0.
Given a linear relation L, one defines the adjoint linear relation L * by L is called symmetric if L ⊂ L * and L is self-adjoint if L = L * . Clearly if B is a closed symmetric linear operator with adjoint B * then the graph, G(B) of B is a closed symmetric linear relation, and the graph, G(B * ) of B * is the adjoint relation to G(B).
In this paper we will be considering closed symmetric linear transformations B with deficiency indices (n, n), which are not necessarily densely defined. If this is the case then this means that B does not have a uniquely defined adjoint operator, and it will be convenient to identify B with its graph G(B): is a closed linear relation but not the graph of a linear operator. Indeed, observe that if g ⊥ Dom(B) then by equation (3.1), (0, g) ∈ G(B) * since for every (f, Bf ) ∈ G(B). For convenience we will simply write B * for G(B) * in the case where B is not densely defined. Note that One can show that if B has deficiency indices (n, n) that the co-dimension of Dom(B) is at most n: and so (1 − V * )f ∈ Ker(V ) which is n−dimensional. Now (1 − V * )f = 0 as then f would be an eigenfunction to eigenvalue 1 and V would not be simple. It follows that the dimension of Dom(B) ⊥ is at most n as otherwise we could find a g ∈ Dom(B) ⊥ such that (1 − V * )g = 0. For Then, as in the case of densely defined B, it follows that for any z ∈ C \ R.
If B is a symmetric linear transformation then one can show, whether or not B is densely defined, that dim (Ker(B * − z)) is constant for z ∈ C ± , so that one can define n ± = dim (Ker(B * − z)) for z ∈ C ± . For lack of a reference, here is an elementary proof of this fact. Proof. Given w ∈ C \ R let P w := projection onto Ker(B * − w) = Ran (B − w) ⊥ , and let Q w := projection onto Ran (B − w) so that Q w = 1 − P w . Now fix w ∈ C \ R. Choose any f ∈ Q w H of unit norm, f = 1. Since f ∈ Ran (B − w), we have that f = (B − w)g for some g ∈ Dom(B). Now B − w is bounded below, an easy calculation shows that for any g ∈ Dom(B): Now choose z in the same half-plane as w and consider: It follows that This implies that Since f was an arbitrary norm one vector in Q w H we conclude that Taking adjoints it follows that we also have For fixed w ∈ C + or C − , this is less than one for all z in a small enough neighbourhood of w.
It follows that for z close enough to w we have so that by [1, Section 34] P z H and P w H have the same dimension. It follows that the dimension of

Herglotz Spaces
In this section we will show that any B ∈ S n is unitarily equivalent to the operator of multiplication by z in a certain space of analytic functions called a Herglotz space. Assume that n < ∞.
4.1. Herglotz Functions. It will be convenient to begin with a brief review of the Nevanlinna-Herglotz representation theory of Herglotz functions on both the unit disk D and the upper half-plane C + . Let g be a C n×n -valued Herlglotz function on D, i.e. an analytic function with non-negative real part. Here C n×n is our notation for the n × n matrices over C. Then by the Herglotz representation theorem there is a unique positive Borel C n×n -valued measure on the unit circle T such that The measure σ determines the Herglotz function g up to an imaginary constant so that We will always impose the normalization condition that b = 0 in this paper. Observe that this means that σ is a probability measure, i.e. σ is unital, σ(T) = 1, if and only if g(0) = 0. We will also extend g to a function on C \ T using the convention that g(1/z) * = −g(z). Now let G := g • b be the corresponding matrix-valued Herglotz function on C + (G has nonnegative real part in C + ). Setting w := b −1 (z) and t = b −1 (α), we obtain that The convention that g(1/z) * = −g(z) implies that G(w) * = −G(w) and this extends G to a function on C\R. Again, we have that σ is unital if and only if g(0) = 1 which happens if and only if G(i) = 1.
Now the Herglotz theorem on the upper half-plane states that for unique Borel measure Σ obeying ∞ −∞ Σ(dt) 1 + t 2 < ∞, and positive constant matrix c ≥ 0 where y = Im (w) and It will be convenient to determine the relationship between the Herglotz measure σ of g and Σ of G := g • b. As above we let is the Poisson kernel on the disk. We can write where σ is the measure on R defined by σ(Ω) A bit of algebra shows that Some more algebra shows that We conclude that Finally this shows how the measures σ and Σ are related: Now let Θ be an arbitrary contractive n × n matrix-valued analytic function on C + . Then is a Herglotz function on C + .
There is a bijective correspondence between C n×n -valued Herglotz functions G on C \ R and C n×n -valued contractive analytic functions Θ on C + defined by The Nevanlinna-Herglotz representation theory can also be used to define a bijective correspondence between C n×n -valued Herglotz functions on C + and a large class of C n×n -positive matrixvalued measures on R. Namely if g is a Herglotz function on the unit disk which obeys the normalization condition of the previous section (no non-zero constant imaginary part), then as discussed above it uniquely determined by a regular, positive C n×n -valued Borel measure on the unit circle T by the formula: It follows that the Herglotz function G := g • b on C + is uniquely determined by the Herglotz measure Σ and the value of σ({1}) by the formula Conversely given any non-negative matrix P ∈ C n×n and positive C n×n matrix-valued Borel measure on R that obeys the condition: for any v, w ∈ C n , there is a unique Herglotz function G on C + that obeys equation (4.3), or equivalently obeys: It follows that there is a bijective correspondence between Herglotz functions G on C + and such pairs (P, Σ), where P ∈ C n×n is positive and Σ is a positive C n×n valued measure obeying the condition (4.4). This in turn implies there is a bijective correspondence between contractive analytic functions Θ on C + and such pairs (P, Σ). Given Θ we will call the corresponding Σ the Herglotz measure of Θ and we will usually denote this by Σ Θ . Similarly σ θ will denote the Herglotz measure of θ := Θ•b −1 . Note that since we assume any Herglotz function g θ obeys our normalization condition (no non-zero imaginary constant part), it follows that σ θ is unital if and only if g θ (0) = 1 = G Θ (1) which happens if and only if θ(0) = 0 = Θ(i).

4.2.
Herglotz spaces. Let Θ be a C n×n −valued contractive analytic function on C + . The Herglotz space, L(Θ) is the abstract reproducing kernel space of analytic C n -valued functions on C \ R with reproducing kernel Namely given any v ∈ C n and f ∈ L(Θ) and w ∈ C \ R, we have that As shown in [4], if Θ is a Livsic characteristic function so that Θ(i) = 0, and the symmetric linear transformation B with characteristic function Θ is densely defined then one can define a closed simple symmetric linear operator Z Θ ∈ S n (L(Θ)) with domain see [4,Theorem 6.3]. Since we do not assume that all of our symmetric linear transformations are densely defined, we will need to extend this slightly: belongs to S n ( L(Θ)).
The proof of this lemma follows from the vector-valued version of [12,Theorem 5], see also [13]. In particular we use the identity valid for all F, G ∈ L(Θ) proven in [12,Theorem 5] for the case n = 1, and easily verified to also hold for the vector-valued case.
Proof. Let S ±i := {F ∈ L(Θ)| F (±i) = 0}. By de Branges' results on Herglotz spaces, if F ∈ S −i then so that the linear transformation V which acts as multiplication by b(z) obeys V : using the identity stated before the proof.
It is not hard to verify that V is closed, and so Z Θ := b −1 (V ) is a well-defined closed symmetric linear transformation. The symmetric linear transformation Z Θ has indices (n, n) since proving that Z Θ is simple. It remains to check that the domain of Z Θ is equal to This proves that F ∈ Dom(Z Θ ) so that D Θ = Dom(Z Θ ).
Lemma 4.4. Let Θ be a contractive analytic function as above. The Livsic characteristic function of Z Θ is a Frostman shift of Θ: Proof. This is a straightforward calculation using the definition of the characteristic function (equations (2.6), (2.7) and (2.8)) and the reproducing kernel for L(Θ). Let {e j } be the standard orthonormal basis of C n . We can choose With this choice of bases, one obtains Recall here that Θ ZΘ (z) = b(z)B(z) −1 A(z).

Now observe that
Using that G Θ (z) * = −G Θ (z) for the Herglotz function G Θ , we also obtain that It follows that Θ(z) : Substituting in our expression for the reproducing kernel K w (z) yields (ignore the factor of −1) We can ignore the factor of −1 since Θ(z) is defined only up to conjugation by fixed unitaries.
Now straightforward algebra shows that Putting these two formulas together yields the Frostman shift formula.
In particular if Θ(i) = 0 then Θ is equal to the Livsic characteristic function of Z Θ , and Theorem 2.1 allows us to conclude: The following example of symmetric extensions of a symmetric operator B with Θ B inner will be important: Example 4.6. Let Θ, Φ be C n×n -valued inner functions on C + such that Θ ≤ Φ. In this case Θ −1 Φ is also an inner function.
Given any inner function Θ one can define a symmetric linear transformation Z Θ acting in K 2 see for example [3,4]. It is straightforward to show that the characteristic function of Z Θ is the Frostman shift of Θ as above so that by Livsic's theorem Z Θ ≃ Z Θ .
It follows that since K 2 Θ ⊂ K 2 Φ that Dom(Z Θ ) ⊂ Dom(Z Φ ) and that Z Θ ⊂ Z Φ so that Z Θ Z Φ . Moreover given any A ∈ Ext(Z Φ ), then the restriction A ′ of A to its smallest invariant subspace containing K 2 Θ belongs to Ext(Z Θ ). This can be generalized further: Suppose that Φ is an arbitrary contractive analytic function such that Φ ≥ Θ where Θ is inner. Then by [14,, K 2 Θ is contained isometrically in the deBranges-Rovnyak space , the operator of multiplication by V (z) is an isometry of K 2 Θ into L(Φ), and by the definition of Dom(Z Θ ), and the definition of this also shows that Z Θ Z Φ whenever Θ is inner, Φ is contractive and Θ ≤ Φ. Again the restriction of any A ∈ Ext(Z Φ ) to its smallest invariant subspace containing V K 2 Θ belongs to Ext U (Z Θ ). Here recall that given B ∈ S , Ext U (B) is the set of all self-adjoint linear transformations A such that A ∈ Ext(U BU * ) for some isometry U : H → K.
We can also construct examples of symmetric B 1 ∈ S n (H 1 ) and B 2 ∈ S m (H 2 ) such that B 1 B 2 where n = m: Suppose that Φ := ΘΓ where Φ, Θ, Γ are all scalar-valued inner functions on C + . Let Then Λ is a 2 × 2 matrix-valued inner function, and note that Z Λ has indices (2,2), and that there is a natural unitary map W from K 2 so that if we view elements of K 2 Λ as column vectors then W acts as multiplication by the 1 × 2 matrix function W (z) = (1, Θ(z)).
Proof. By Corollary 4.5, B j ≃ Z Θj . As discussed in the above example if Θ 1 is inner and Θ 1 ≤ Θ 2 then Z Θ1 Z Θ2 so that B 1 B 2 .
Given any B ∈ S 1 (H), it is well known that there is a conjugation C B which commutes with B, i.e. C B : Dom(B) → Dom(B) and C B B = BC B . Recall here that a conjugation is an anti-linear, idempotent onto isometry [5, Theorem 7.1]. It will be useful for us to extend this construction to the case of arbitrary B ∈ S . We say that C is a conjugation intertwining B 1 ∈ S n (H 1 ) and B 2 ∈ S n (H 2 ) provided that CB 1 = B 2 C, and C is an anti-linear and onto isometry. Proposition 4.8. Let Θ be a contractive C n×n -valued analytic function in C + , n ∈ N. The map In the above T denotes matrix transpose and for a vector F (z), F (z) denotes the vector obtained by taking the complex conjugate of each component in the fixed canonical basis of C n .
Proof. Let {e k } denote the canonical orthonormal basis of C n . Let C : C n → C n denote the conjugation defined by entrywise complex conjugation: if v = c i e i for c i ∈ C, then C v := c i e i . Given any matrix A ∈ C n×n , with entries A = [a ij ], it is easy to check that CAC = [a ij ] = (A * ) T = (A T ) * . By definition, given F ∈ L(Θ), we have that The closed linear span of the evaluation vectors K Θ w v for w ∈ C \ R, v ∈ C n is dense in L(Θ). The action of C Θ on such functions is This proves that , and it follows from the density of the point evaluation vectors that C Θ : L(Θ) → L(Θ T ), and that it has dense range. It is clear by definition that C Θ is anti-linear. To see that it is an (anti-linear) isometry note that Using the fact that linear combinations of such functions are dense in L(Θ) and L(Θ T ), we conclude that C Θ is an isometry with dense range, and hence is onto. In other words, C Θ is anti-unitary, so that C * Θ C Θ = 1. As is easy to check: Finally, since Dom(Z Θ ) := {F ∈ L(Θ)| zF ∈ L(Θ)}, and similarly for so that C Θ F ∈ Dom(Z Θ T ), and conversely given any G ∈ Dom(Z Θ T ), C Θ T G ∈ Dom(Z Θ ), and C Θ C Θ T G = G, showing that C Θ Dom(Z Θ ) = Dom(Z Θ T ). The above arguments also show that for completing the proof.
Note that any such conjugation Composing the unitary operators effecting these equivalences with C Θ yields C B .
4.10. Measure spaces. Let Σ be any C n×n positive regular matrix-valued measure on R which obeys the Herglotz condition: for any v, w ∈ C n . We define the measure space L 2 Σ to be the space of all C n -valued functions on R which are square-integrable with respect to Σ, i.e. f ∈ L 2 Σ provided that for any z ∈ C \ R, define the C n×n matrix function Suppose that Θ is a contractive analytic function such that The deBranges isometry and the orthogonal complement of the range of W Θ is the closed linear span of the constant functions P C n . One can then check that the reproducing kernel for L(Θ) is given by the formula Also notice that if P = 0 and Θ is a characteristic function so that Θ(i) = 0, that this implies that G Θ (i) = 1 so that and this implies that the vectors δ i e k , 1 ≤ k ≤ n are an orthonormal set.

Non-canonical representations of symmetric operators
We are now sufficiently prepared to begin pursuing the main theory and results of this paper. For any A ∈ Ext(B) we can construct a representation of B as multiplication on a space of analytic functions on C \ R as follows: These two formulas coincide when U does not have 1 as an eigenvalue.
Then it is not difficult to verify as in [5, Section 1.2] that (regardless of whether A is densely defined or not) for any w, z ∈ C \ R, U w,z has the following properties: Given any fixed w ∈ C \ R, let J w : C n → Ker(B * − w) be a bounded isomorphism (a bounded linear map with bounded inverse). We can then define the map (the last formula holds for the case where A is densely defined) where P w projects onto Ker(B * − w) and it follows that if A ∈ Ext(B) is actually a canonical element of Ext(B) that Γ A is a model for B as defined in [4]. Namely, recall: , the space of bounded linear maps from J to H, is a model for B if Γ satisfies the following conditions: where denotes the closed linear span.
Recall that as shown in [4], any model Γ for B ∈ S n (H) can be used to construct a reproducing kernel Hilbert space of analytic functions H Γ on C \ R and a unitary U Γ : H → H Γ such that the image of B under this unitary transformation acts as multiplication by z.

Now if A ∈ Ext(B) is non-canonical then Γ w
A as defined in equation (5.3) does not necessarily satisfy the conditions of a model as defined in Definition 5.1. Despite this Γ w A has similar properties to a model and can still be used to construct a reproducing kernel Hilbert space of analytic functions H A on C \ R, and (at least in the case under consideration where Θ B is inner) an isometry U A : H → H A such that U A BU * A again acts as multiplication by z in H A . This motivates the definition of a non-canonical model which includes these generalized models Γ A arising from non-canonical A ∈ Ext(B): H) is a quasi-model for B ∈ S n (H) if Γ satisfies the following two conditions: Given a quasi-model Γ, we define
We will also use the notation We will now show that any quasi-model Γ of rank (n, n) has a property similar to the property (5.6) for a model.
is a quasi-affinity on J whenever m + = n and z, w ∈ Π + Γ or whenever m − = n and z, w ∈ Π − Γ .
Remark 5.6. Note that in the case where n < ∞, which is the case we are primarily studying, when Γ(z) * Γ(w) is a quasi-affinity, it is acting between finite dimensional spaces and hence is in fact bounded and invertible. Also the reason this proposition is important is that we will shortly construct a reproducing kernel Hilbert space H Γ whose reproducing kernel is K w (z) = Γ * (z)Γ(w), and it will be useful to know when this is invertible.
This proposition will be the consequence of the following: is a quasi-affinity for any z, w ∈ C + or z, w ∈ C − , i.e. it is injective and has dense range (and hence an inverse which is potentially unbounded).

The proof of this proposition needs a little set up. Given a closed linear transformation
and ∔ denotes the non-orthogonal direct sum of linearly independent subspaces.
Proof. It suffices to prove that G z is the graph of a densely defined closed linear operator.
Clearly G(B z ) ⊂ B * . To prove that G(B z ) is the graph of a linear transformation, we need to prove that the intersection of the multi-valued part of B * with G z is the zero element: Suppose not, then we can find a sequence (f n ) ⊂ Dom(B) and a sequence h n ∈ Ker(B * − z) such that (f n + h n , Bf n + zh n ) → (0, g) where g ⊥ Dom(B). It follows that However this would then imply that which is impossible as B is symmetric and z ∈ C \ R. This proves that G z is the graph of a linear transformation B z , it remains to prove that B z is densely defined.
To prove that B z is a linear operator, i.e. densely defined, suppose that φ ∈ H is orthogonal to Dom(B z ). Then φ ⊥ Dom(B) and φ ⊥ Ker(B * − z). Hence φ ∈ Ran (B − z) and so φ = (B − z)f for some f ∈ Dom(B). But φ is orthogonal to Dom(B) as well so that which as before is impossible as B is symmetric.
Since B z is a closed linear operator, the proof is identical to that of [4, Lemma 2.6], and we omit it.
Proof. (of Proposition 5.7) Given a unit vector c ∈ C n (we take C ∞ := ℓ 2 (N)), let . But then, since w does not belong to the spectrum of B z , which shows that ψ z ∈ Dom(B), contradicting the fact that B is symmetric.
is also injective, proving that Y (w, z) also always has dense range. This proves that Y (z, w) is always a quasi-affinity of B(ℓ 2 (N)) whenever z, w are both in C + or are both in C − .

Proof. (of Proposition 5.5)
If z, w ∈ Π + Γ this follows from the observation that given any orthonormal basis {j k } of J , and z ∈ Π + Γ , δ k (z) := Γ(z)j i forms a basis for Ker(B * − z), and that The proof of the other half of the proposition is analogous.
For the remainder of this section we will assume that n < ∞, although many of our arguments generalize to the case n = ∞ without too much difficulty.
Lemma 5.10. The sets Σ ± Γ = C ± \ Π ± Γ are contained in the zero-sets of non-zero analytic functions in C ± (and hence are purely discrete with accumulation points lying only on R ∪ {∞}).
Proof. Choose any w ∈ Π + Γ . Let {j k } be an orthonormal basis of J such that {j k } m+ k=1 is an orthonormal basis of Ker(Γ(w)) ⊥ . Let {v k } m+ k=1 be the basis of Ran (Γ(w)) defined by v k = Γ(w)j k and set and let δ w (z) := det D w (z). Then δ w is analytic (as a function of z) in C + and δ w is not identically zero since δ w (w) = det D w (w), and it is clear that by construction D w (w) is invertible. Now if z ∈ C + is any point such that δ w (z) = 0 then D w (z) is invertible and hence Γ(z)| Ker(Γ(w)) ⊥ is invertible as a map onto its range. Let j k := P z j k where P z projects onto Ker(Γ(z)) ⊥ . The j k form a linearly independent set since otherwise the set of all would not be linearly independent, contradicting the fact that for any z ∈ C + such that δ w (z) = 0, proving the claim.
Corollary 5.11. Given any w ∈ Π ± Γ we have that the set is contained in the zero set of an analytic function which is not identically zero.
Proof. This is intuitively clear. Since B is simple, z∈C\R Ker(B * − z) is dense in H. By definition if z / ∈ Σ + Γ and m + = n then Ker(B * − z) = Γ(z)J . By Lemma 5.10 the set Π + Γ of all z ∈ C + for which Γ(z) is invertible is dense in C + .
If f ∈ H and f ⊥ z∈C+ Γ(z)J then f ⊥ Ker(B * − z) for all z ∈ Π + Γ . Let Γ be a canonical model for B, and let f (z) := Γ(z) * f . Since f ⊥ Ker(B * − z) for all z ∈ Π + Γ , the J -valued analytic function f (z) vanishes everywhere on Π + Γ . Since this set is dense in C + , f = 0 identically on C + . This shows that f ⊥ z∈C+ Ker(B * − z). The same argument in C − completes the proof.
Definition 5.13. We say that a quasi-model Γ is a generalized or non-canonical model for B if z∈C\R Ran (Γ(z)) = H. By Lemma 5.12, any full rank quasi-model (a rank (n, n) quasi-model) is a generalized model for B. The next proposition verifies that the linear maps Γ w A defined for A ∈ Ext(B) and w ∈ C \ R in equation (5.3) satisfy our definition of a quasi-model. Proposition 5.14. If B ∈ S n (H) with Θ B inner and A ∈ Ext(B), then for any w ∈ C \ R one can construct a generalized model Γ w A for B by defining J := C n , J w : J → Ker(B * − w) a bounded isomorphism and letting We will usually assume that J w is chosen to be an isometry. Recall that if A is such that (5.1). In the exceptional case where 1 ∈ σ p (U ), U wz is given by equation (5.2).
(as discussed at the beginning of this section).
Note that by construction Γ w A (w) = J w , which is invertible by assumption. Given any A ∈ Ext(B), we are free to choose w ∈ C \ R in the construction of a quasi-model Γ w A associated with A. For the remainder of this paper we will choose w = −i unless otherwise specified and define , which (excluding the exceptional case) is equal to and Γ A (i) = J −i . We will also simply write J for J −i where J = P −i J : C n → Ker(B * + i), and usually we assume J is an isometry.
Remark 5.15. Suppose that the characteristic function Θ of B is inner. If this is the case then we will show that for any A ∈ Ext(B), that z∈C− Ran (Γ w A (z)) = H for any w ∈ C + and z∈C+ Ran (Γ w A (z)) = H whenever w ∈ C − . To see this note that in this case that B is unitarily equivalent to Z Θ , which acts as multiplication by z in some model space where T denotes transpose, as defined in [15,Claim 3]. The existence of C Θ also follows from our Corollary 4.9.
Suppose that w ∈ C + . Then by Lemma 5.12, since m − = n for any Γ w This proves that if Θ B is inner, then every quasi-model Γ w A for A ∈ Ext(B) and w ∈ C \ R is a generalized model.
Suppose that B ∈ S n (H) and Θ B is inner so that for any A ∈ Ext(B) and w ∈ C \ R, Γ w A is a generalized model for B (see Remark 5.15 above).
Let V := the partial isometric extension of b(B) to H. Given any C ∈ B 1 (C n×n ) definê a contractive extension of V and let (U C , K) be the minimal unitary dilation of V (C). Choose C = 1 m where 0 ≤ m < n so thatĈu j = 0 for any n ≥ j > m.
for any n ≥ j > m. Now given any z ∈ C + , Since both z, i ∈ C + , by dilation theory this is just equal to so that Γ C (z)e j = 0 for any n ≥ j > m as before. To see this note that since Given any z ∈ C − that lies in the open ball of radius 1 about z = −i we have that (A C − z) −1 can be expressed as a power series in (A C + i) −1 , and it follows from this that the resolvent formula: Hence Γ(z)u j is identically zero in C + for any n ≥ j > n so that Similarly using Γ = Γ −i AC instead, one can construct an example of Z A with indices (n, n) where n > m − .

Construction of the model reproducing kernel Hilbert space
Given any quasi-model Γ for B ∈ S n (H), we can construct a reproducing kernel Hilbert space H Γ as follows: an analytic function on C \ R, and let H Γ := the vector space of all the functionsf . Let We say that the reproducing kernel Hilbert space H Γ has the division property in Proposition 6.2. With the above inner product H Γ is a reproducing kernel Hilbert space of analytic functions on C \ R with reproducing kernel k Γ w (z) = Γ(z) * Γ(w), and point evaluation vectors k Γ w j = U Γ Γ(w)j, for j ∈ J . If the rank of Γ is (m + , m − ) then H Γ has the division property in C ± whenever m ± = n.
The map U Γ : H → H Γ defined by U Γ f =f is a co-isometry with initial space H −1 Γ , and is unitary if and only if Γ is a generalized model for B. If Γ is a generalized model then Z Γ := U Γ BU −1 Γ acts as multiplication by z on the domain U Γ Dom(B), and if either m + or m − is equal to n then Recall that if Θ B is inner then given any A ∈ Ext(B), and w ∈ C \ R, any quasi-model Γ w A is a generalized model with indices (m + , n) or (n, m − ).
Proof. This is all fairly straightforward to check. First of all one should verify that f Γ = 0 implies that f ⊥ H −1 Γ , i.e. thatf (z) = Γ(z) * f = 0 for all z ∈ C \ R. Indeed Γ(z) * f = 0 for z ∈ C \ R if and only if f, Γ(z)j = 0 for all z ∈ C \ R, and j ∈ J which happens if and only if f ⊥ Ran (Γ(z)) for all z ∈ C \ R, in other words f ⊥ H −1 Γ . Now given any j ∈ J and f ∈ H −1 and it follows from this that for any j ∈ J, k w j := U Γ Γ(w)j are reproducing kernel vectors in H Γ and the reproducing kernel is given by Let us first show that Z Γ acts as multiplication by z on its domain. If f ∈ Dom(B) then showing that Z Γf (z) = zf (z). Now suppose that m + = n, and let's prove that H Γ has the division property in C + . In this case iff ∈ H Γ andf (w) = 0 then for any j ∈ J. If w ∈ Π + Γ , then Γ(w) : J → Ker(B * − w) is onto which implies that f ∈ Ran (B − w).
6.3. Alternate formulas for the Livsic characteristic function. In this subsection we pause to compute an alternate formula for the Livsic characteristic function. This will be useful, in particular, for computing formulas for the reproducing kernel of H Γ in the next subsection.
Suppose that B ∈ S n (H) where n < ∞. As mentioned in the introduction the Livsic characteristic function of B is usually defined using and Here is an alternate formula that is sometimes useful. Let A be a canonical self-adjoint extension of B and let w j (z) : is the standard orthonormal basis of C n , and J i is an isometry.
Then it follows that This shows that and a similar calculation shows that It follows that A(z) * = B(z).
6.4. Reproducing Kernel formulas for H Γ . Let Γ be a generalized model for B of rank (m + , m − ) where at least one of m ± is equal to n. Then by Proposition 6.2 we have an isometry so that Z Γ is unitarily equivalent to B.
For any w ∈ C \ R let P w be the projection onto Ran (B − w) = Ker(B * − w) ⊥ , and let Q w := U Γ P w U * Γ , the projection onto Ran (Z Γ − w). Now define and that L * w = L w . We can now calculate formulas for the reproducing kernel of H Γ , using the same procedure as in [4,Section 4]. Let k w (z) = k Γ w (z) denote the reproducing kernel of H Γ . Now given any u, v ∈ J and α ∈ C \ R, But also, Solving for k w (z)u, v and using that u, v ∈ J were arbitrary yields: for any α ∈ C \ R.
Now suppose Γ is a rank (m − , n) quasi-model for B and that Z Γ is unitarily equivalent to B. This happens for example if Γ = Γ A for some A ∈ Ext(B). Also choose J := C n , and J : C n → Ker(B * + i) to be an isometry, and α = i in equation (6.7).
Let {u k } be an orthonormal basis for Ker(B * − i) such that {u k } n+ k=1 is a basis for Ran (Γ(−i)), and let {v k } n k=1 be an orthonormal basis for Ker(B * + i) such that v k = Je k , and {e k } is an orthonormal basis of C n . We assume here that J is an isometry.
Using that 1 − Q −i = n l=1 ·,û l û l we can compute: Hence the above can be written: Compare this to This proves that A similar calculation shows that since (1 − Q i ) = n l=1 ·,v l v l we get that If we take β(z) := [ v l , w k (z) ] then as before it is not hard to check that It follows that our formula for the reproducing kernel in H Γ can be written: Now in the case where both z, w ∈ Π Γ (in particular for Γ A we have that Π + A is dense in C + ) we have that {w j (z)} and {w j (w)} are bases for Ker(B * − z) and Ker(B * − w) respectively, so that for such z, w we have β = B and α = A, where A, B are the matrices in the definition of Θ B (see

Hence for any
Also observe that by the formula (6.11) we have that

Cyclicity
The goal of this section is to show that the characteristic function Θ B of B is inner implies that Ker(B * − w) is cyclic for any A ∈ Ext(B). This will enable us, in the subsequent section, to extend the isometry U A : H → H A to an isometry V A : K → K A where A is self-adjoint in K, K A ⊃ H A is a larger reproducing kernel Hilbert space on C \ R containing H A , and V A | HA = U A . This larger space K A contains information about the extension A ∈ Ext(B) that will be key for our characterization of Ext(B).
Here we say that a subspace S ⊂ H is cyclic for A ∈ Ext(B) if It will be convenient to apply some of the dilation theory for contractions as developed in [7]. The tools we are going to use are described below: Given a contraction T ∈ B(H), recall that the defect indices of T are defined to be the pair of positive integers (d T , d T * ) where Given B ∈ S 1 (H), we will be studying the contraction This is a partial isometry, and it is clear that the defect indices of V are equal to the deficiency indices of b w (B), namely (n, n). A contraction is called c.n.u. (completely non-unitary) if it has no non-trivial unitary restriction. It is clear that since B is simple, this implies that V is c.n.u. The model theory of Nagy-Foias [7] associates a contractive operator-valued function Θ T called the Nagy-Foias characteristic function of T , to any c.n.u. contraction T . This function is defined by In our case where T = V is a partial isometry, this expression simplifies to: Since Θ V coincides with the Livsic characteristic function of b w (B) (this is a consequence of the fact that V is a partial isometry, as discussed above), we conclude that Θ B is inner if and only if both V k → 0 strongly and (V * ) k → 0 strongly. In the notation of [7], if Θ B is inner so that V k → 0 and (V * ) k → 0 strongly, and V has defect indices (n, n), V is called a contraction of class C 0 (n).
By [7, II.3.1] the projection P * onto R * can be calculated by the formula: . Then for any h ∈ H, Lemma 7.2. Let B ∈ S n (H), A ∈ Ext(B). Given any h ∈ H then for any k ∈ N: In the above recall that b w (z) : Proof. This clearly holds if h ∈ Ran (B − w) Repeating this process k times yields h 2k+1 = (b w (B)Q w ) k+1 and one obtains the formula stated above, namely,

Proof. (Theorem 7.1) In the case where
Now we use the formula (7.1) of Nagy-Foias to conclude that if A = b −1 w (U ) where U is the minimal unitary dilation of V , that U −(k+1) V k+1 h → P * h, proving the formula (7.2) and the theorem.  Proof. By Lemma 7.2, given any A ∈ Ext(B) and h ∈ H, Hence to prove the formula (7.9), it suffices to show that and so the formula (7.9) holds. If A = b −1 w (U ), then the fact that Ker(B * − w) is cyclic follows from the fact that R * = {0}. For arbitrary A ∈ Ext(B), the formula (7.9) shows that the cyclic subspace S w for any fixed A ∈ Ext(B) generated by Ker(B * − w) contains H. Hence if A is self-adjoint in K, then S w = K, as otherwise K ⊖ S w would be a non-trivial subspace of K ⊖ H which is reducing for A (this contradicts one of our assumptions on Ext(B)). This proves that Ker(B * − w) is cyclic for any A ∈ Ext(B).

Conversely if Ker(B * − w) is cyclic for any A ∈ Ext(B), then it is cyclic for
where U is the minimal unitary dilation of V = b w (B)Q w , and it follows from the definition of R * that P * = 0, and hence T n → 0 strongly. If n < ∞ this implies T is a contraction of class C 0 (n), implying that the characteristic function Θ B of B is inner as discussed previously. If n = ∞ our assumption that Ker(B * − w) is cyclic also implies that (T * ) k → 0 strongly as well so that we get that Θ B is inner.
Note that the above proof also shows: Corollary 7.5. If B ∈ S n (H), n < ∞, and there is a w ∈ C \ R such that Ker(B * − w) is cyclic for every A ∈ Ext(B), then Θ B is inner. where recall that provided A = b −1 (U ) and U does not have 1 as an eigenvalue then

A larger reproducing kernel Hilbert space K A ⊃ H
where recall that J = J −i = P −i J −i and J : C n → Ker(B * + i). In the exceptional case where A ∈ Ext(B) is defined using a unitary extension U of b(B) and 1 ∈ σ p (U ), recall that U w,z is given by formula (5.2). We will assume in this section that J is an isometry. Note that We define a new reproducing kernel Hilbert space K A as the abstract C n -valued reproducing kernel Hilbert space on C \ R with reproducing kernel K w (z) := Ω(z) * Ω(w) The existence of K A follows from the fact that K w (z) is a positive kernel function, and the abstract theory of reproducing kernel Hilbert spaces [18,Theorem 10.11].
Observe that the difference is a positive kernel function. The theory of reproducing kernel Hilbert spaces then implies that H A is contractively contained in K A [18, Theorem 10.20].
For v ∈ C n the function K w v defined by

Now define a linear map V
Note that if g ∈ H that so that for any g ∈ H, Hence if E A : H A → K A is the contractive embedding then Also observe that if u ∈ C n , then is the u point-evaluation vector in K A at w.
Proof. Recall that since we assume that B is such that Θ B is inner, Corollary 7.3 implies that Ker(B * + i) is cyclic for A.
Since Ker(B * + i) is cyclic, K is spanned by vectors of the form Ω(w)J v for w ∈ C\ R and v ∈ C n . In particular for any v ∈ C n , the vector To see that V A is an isometry use that vectors of the form f = j c j Ω(w j ) v j , for v j ∈ C n and w j ∈ C are dense in K, so that Hence the contractive embedding E A : H A → K A is actually an isometric inclusion, and H A ⊂ K A as a Hilbert subspace. In more detail, if φ := φ[U ; V ], then

Cauchy transforms and characteristic functions for A ∈ Ext(B). For any
Equivalently if we impose the normalization condition discussed in Section 4, By the relationship between Herglotz functions on the disc and upper half-plane, as discussed in Section 4, we have that or equivalently In particular if U does not have 1 as an eigenvalue, then Φ A is uniquely determined by Σ A . Note that since U is unitary, the projection-valued measure P U is unital which implies that σ U is a unital probability measure so that g φ (0) = 1, and this in turn implies that φ(0) = 0, and that Namely in [19], Donoghue defines the Weyl-Titchmarsh function of a pair (B, A), where B is a densely defined simple symmetric operator with deficiency indices (1, 1) and A ∈ Ext(B) by the formula where g + is a fixed normalized element in Ker(B * − i). In this case where B has indices (1, 1), we can define our isometry J : C → Ker(B * + i) in the construction of Γ A and Ω A by Je 1 = g − where e 1 = 1 is a trivial orthonormal basis of C and g − is a fixed unit element of Ker(B * + i). In this case the Herglotz function G ΦA is just In [20], the Livsic function of the pair (A, B), where B as above has indices (1, 1) is defined to be Now let {u j }, {v j } be orthonormal bases of Ker(B * − i) and Ker(B * + i) respectively, and leť u j = C B u j andv j = C B v j be corresponding basis elements for Ker(B * T ± i), and suppose that J ±i : C n → Ker(B * ∓ i) are isometries defined by J −i e k = v k , J i e k = u k , and defineJ ±i similarly. Then C B J ±i =J ±i , and it follows that if where Q A (Ω) := P H P A (Ω)P H , then this is the suitable generalization of Donoghue's Weyl-Titchmarsh function to the case where B has indices (n, n) and is not necessarily densely defined. Moreover  Proof.
, where the last equality follows from equation (4.5) and the definition of Φ A . Now let us compute the Livsic characteristic function of the operator Z := Z ΦA ∈ S n ( L( Φ A )) which acts as multiplication by the independent variable in L( Φ A ) = K A . We have Note that K i (i) = K −i (−i) = J * J = 1. By Section 6.3, we can compute the Livsic characteristic function of Z in two ways: We have that Similarly and Livsic's theorem implies that the functions are both contractive and equal (modulo multiplication to the left and right by fixed unitaries) to the Livsic characteristic function Θ Z of Z. Recall that the Livsic characteristic function is only defined up to unitary coincidence, so this means that there are fixed unitary matrices U, V such that Explicitly we have Proof. Since Λ A is the characteristic function of Z ΦA , and since K ΦA . This shows that Λ A is the Livsic characteristic function of Z ΦA , and Lemma 4.4 of Section 4 implies that Λ A is the Frostman shift of Φ A which vanishes at i. However since Φ A (i) = 0, this Frostman shift is just equal to Φ A and Φ A = Λ A .
Proof. Consider again the symmetric linear transformation Z which acts as multiplication by z in L( Φ A ) = K A , where Φ is the contractive analytic function corresponding to the measure πΣ A . We can construct a canonical model for Z by choosing J := C n with orthonormal basis {e j } and defining where K z (w) is the reproducing kernel for K A . If we do this we find that (K A ) Γ = K A and that U Γ is just the identity on K A . Hence it follows from Section 6.4, and in fact from [4], that we can express the reproducing kernel for K A as We can write this as Also note that if k w (z) is the reproducing kernel for H A , then for any z ∈ Π + A (which is dense in proving the theorem. This is a unitary matrix acting on C 3 , and U | Ker Our goal is to calculate Φ A and to verify that Φ A ≥ Θ B . Note that to avoid writing column vectors we will write (a, b) T to denote the transpose of the row vector (a, b), and sometimes we will omit the T in our calculations.
To calculate the Livsic characteristic function we also need to determine Ker(B * − z). First we calculate Ran (B − z): Since Ker(V ) ⊥ is spanned by e 1 and V e 1 = e 2 , we get that Ran (B − z) is spanned by and this shows that Ker(B * − z) is spanned by To calculate Φ A , we first need to calculate the projection-valued measure of U . We begin by calculating the eigenvalues and eigenvectors of U : We have where λ 1 = 1, λ 2 := −4/5 + i3/5 =: β and λ 3 = λ 2 = β. A normalized eigenvector for λ 1 = 1 is: It follows that the projection-valued measure of U is given by where the δ λi are Dirac point measures of weight one at the points λ i , and theb i are normalized eigenvectors to the eigenvalues λ i . Now the scalar measure where v = e 1 is a unit vector spanning Ran (V ) ⊥ = Ker(B * + i). Hence Now e 1 ,b 1 = 2/3, and since b 2 = C b 3 is the component-wise complex conjugate of b 3 , it follows that | e 1 ,b 2 | 2 = | e 1 ,b 3 | 2 =: a. Finally since P U is unital, σ U must be a probability measure: proving that a = 5/18. In conclusion, where β = −4/5 + i3/5. It follows that where σ U := σ U • b. An easy calculation shows that b −1 (β) = 1/3 and b −1 (β) = −1/3, and so it follows that Notice that G ΦA (i) = 1 as expected. Hence .
The numerator simplifies to Let p(z) = n(z) −9 = z 3 − i 9 4 z 2 − 3 2 z + i 4 . It follows that Φ A (z) is the product of three Blaschke factors, one for each of the roots of p(z). It is easy to calculate that p(z) factors as p(z) = (z − i) 2 (z − i 4 ), and so (up to a unimodular constant), which is indeed greater or equal to Definition 8.11. We say that Suppose that A 1 , A 2 ∈ Ext(B) are such that A k = b −1 (U k ) for U k ∈ Ext(b(B)) which do not have 1 as an eigenvalue. Then: Theorem 8.12. A 1 ∼ A 2 if and only if A 1 ≃ A 2 via a unitary U whose restriction to H is the identity.
The above result is easily extended to include the exceptional case where one (or both) A 1 , A 2 are is a positive operator-valued measure acting on H. Assume for now that A is defined using a W ∈ Ext(U b(B)U * ) which does not have 1 as an eigenvalue so that A = b −1 (W ).
Since A = b −1 (W ) and 1 / ∈ σ p (W ), A is a densely defined self-adjoint operator and P A (R) := χ R (A) = 1 K , as otherwise P A (R)K is a non-trivial reducing subspace for A which contains H. In other words the projection-valued measure of A is unital. By Naimark's dilation theorem there is a larger Hilbert space K ′ ⊃ H and a unital projectionvalued measure P (Ω) acting on K ′ such that the compression P H P (Ω)P H = Q(Ω), for any Borel set Ω. This projection-valued measure P is called a dilation of Q, and it can be chosen to be minimal in the sense that K ′ = P (Ω)H. If A ′ is the self-adjoint operator corresponding to this projection valued measure, then it follows that A ′ ∈ Ext(B). It is also clear that by definition, If A is defined using W ∈ Ext(U b(B)U * ) with 1 ∈ σ p (U ), define Q(Ω) = P H P U (Ω)P H , a unital positive operator-valued measure (POVM) on the unit circle. Again apply Naimark's dilation theorem to obtain a unitary operator U ′ on K ′ ⊃ H. As before it follows that if Recall that the Alexandrov-Clark measures for θ V are defined as the n×n matrix-valued measures δ U for any U ∈ U(n) (the group of n × n unitary matrices) associated with the Herglotz functions g U := 1 + θ V U * 1 − θ V U * , via the Herglotz representation theorem for the unit disk i.e.
be the corresponding Herglotz function on C + . We define the Alexandrov-Clark measures of Θ B to be the measures ∆ U on R such that Recall that as discussed in Section 4 (see equation (4.1)) we have that Now let Z denote the unitary operator of multiplication by z in L 2 θ (T) (the L 2 space of vectorvalued functions on T which are square integrable with respect to the measure δ 1 ).
Let {b − j (z) = e j } be a basis for the constant functions in L 2 θ . Since Θ(i) = 0 = θ(0), it follows that this is an orthonormal basis. Similarly define b + j (z) := 1 z e j . For any A ∈ C n×n let Z(A) where P − projects onto the closed span of the b − j , and A = j * Aj where j is an isomorphism defined by je k = b − k which takes C n onto the range of P − . Then as shown in [8] Z(0) has Livsic characteristic function θ V , and so it follows that there is a unitary transformation W : H → L 2 θ that implements the equivalences Z(0) ≃ V = b(B)(1 − P i ), and Z(U ) ≃ V (U ) for any U ∈ U(n), and such that W : [8,2], where {u j } is an orthonormal basis of Ker(B * − i).
Moreover the results of [2] show that Using the fact that G U = g U • b, and the relationship between Herglotz functions and measures on the upper half-plane and the disk as described in Section 4, it follows that and where S ±i = P ±i H, and P ±i are the projections onto Ker(B * T ± i). Similarly define S ±i and P ±i . As above, U is defined by Given any g ∈ Dom(B T (U )), it follows that there is some f = f , u i u i ∈ S i and g T ∈ Dom(B T ) such that , and where

Comparing this toÛ
1−z , so that we also have that Dom(B(R)) = Ran (1 − V (R) * ). It follows that and this proves that It further follows that Now let δ U be the Alexandrov-Clark measure associated with the Herglotz functions where θ T := Θ T • b, and as before let let ∆ U := δ U • b −1 . As discussed before this proof, the results of [2] show that Similarly, In conclusion we have that if Φ : This proves that Φ B(U T ) = (U T ) * Θ B , or equivalently that Φ B(U) = U * Θ B .
Proof. (of Theorem 8.14) This map is automatically injective by the definition of ext(B). To show that it is surjective, let Φ be a contractive analytic function such that Φ ≥ Θ B , i.e. Θ −1 B Φ is a contractive analytic function. Let Θ := Θ B . Now we have B ≃ Z Θ acting in L(Θ), and by Corollary 4.5, Z Θ Z Φ . Furthermore by Theorem 8.15 we have that there is a canonical self-adjoint extension A of Z Φ whose characteristic function Φ A = Φ[A; Z Φ ] relative to Z Φ is Φ. Moreover one can see from Example 4.6 that the isometry To see this note that since Θ(i) = 0 = Φ(i) that is an isometry of L(Θ) onto K 2 Θ , that K 2 Θ is isometrically contained in K 2 Φ (since Θ is inner), and that multiplication by , This shows that the characteristic function Φ A = Φ A [A; Z Φ ] = Φ of A with respect to Z Φ is the same as the characteristic function Φ A of A ∈ Ext U (B) with respect to B. By Remark 8.13, there is an Putting it all together we have that This proves surjectivity.

Partial order calculations
In this section we study the partial order on symmetric linear transformations described in the introduction: Recall here ≃ denotes unitary equivalence.
We assume in this section that n < ∞, and under this assumption, it is not difficult to verify that is indeed a partial order on the unitary equivalence classes of S (see [22]). Also, using the Cayley transform, this also defines a partial order on V . Namely, given This is the same, modulo unitary equivalence as the partial order defined on partial isometries by Halmos and McLaughlin in [6]. That is, they define The main goal of this section is, given B 1 , B 2 ∈ S with Θ 1 := Θ B1 inner, to provide necessary and sufficient conditions on the characteristic function Θ 2 := Θ B2 of B 2 so that B 1 B 2 .
Let B 1 ∈ S m (H 1 ) and B 2 ∈ S n (H 2 ) be symmetric linear transformations, and suppose that B 1 B 2 . As always in this paper we assume that Θ 1 is inner.
Let Σ := πΣ 2 where Σ 2 is the Herglotz measure on R corresponding to Θ 2 . By Remark 9.2, we can and do assume that B 2 = M Σ .y Recall here that M Σ is the symmetric operator of multiplication by the independent variable in L 2 Σ , where L 2 Σ is the Hilbert space of column vector-valued functions f which are square integrable with respect to Σ, i.e. if f, g ∈ L 2 Σ then Let {e k } be the standard basis of C n , and let {v k } n k=1 be a fixed orthonormal basis of Ker(B * 2 + i). Define an isomorphism J : C n → Ker(B * 2 + i) by Je k = v k . By our previous result, Theorem 8.15, on Alexandrov-Clark measures, there is a canonical A ∈ Ext(B 2 ) such that Since we assume that B 2 = M Σ it actually follows that A = M Σ , the self-adjoint operator of multiplication by t in L 2 Σ .
this follows because if Φ is the contractive analytic function corresponding to Σ, then the deBranges-Cauchy transform isometry, W : L 2 Σ → L( Φ), is onto and acts as and the e j -point evaluation vector at z = i in L( Φ) is Since K i e j spans Ker(Z * Φ + i), the W * K i e j = v j span Ker(M * Σ + i). Moreover this choice of v k defines an orthonormal basis since Θ 2 (i) = 0 implies that It follows from this formula that v k , v j = δ kj .
We are also free to assume that B 1 ⊂ B 2 = M Σ so that B 1 ∈ S m (S) where S ⊂ L 2 Σ . Let A be the restriction of A = M Σ to the intersection of its domain with its smallest reducing subspace containing S. Then A ∈ Ext(B 1 ). Let { v k } m k=1 be an orthonormal basis of Ker(B * 1 + i), and let J : C m → Ker(B * 1 + i) be an isomorphism defined by Je k = v k . Then if we define then we have that Φ := Φ[A; B 1 ] is the contractive analytic function corresponding to Σ ′ and we define Σ = π Σ ′ . Now since {v k } is a cyclic set for M Σ = A , we have that for certain functions D jk where 1 ≤ j ≤ m and 1 ≤ k ≤ n and 1 t+i (D j1 , ..., D jn ) T ∈ L 2 Σ for 1 ≤ j ≤ m. Here the superscript T denotes transpose (we view elements of L 2 Σ as column vector functions).
( Σ(dt)e 1 , e n ) · · · ( Σ(dt)e n , e n ) Hence we have that and using the relationship (9.1) between the v j and the v k we get that where Now since A ∈ Ext(B 1 ), B 1 ∈ S m (S), let K := the cyclic subspace of L 2 Σ generated by S and A. Let U : L 2 Σ → L 2 Σ be defined by multiplication by D(t), V A : K → K A be the model space isometry, and W : L 2 Σ → K A be the deBranges Cauchy transform isometry.
Claim 9.4. The linear map U : L 2 Σ → L 2 Σ is an isometry which obeys V A U = W, and U U * = 1 K .
Since U acts as multiplication by the matrix function D(t), it is easy to see that U will intertwine M Σ and M Σ . However we also want to verify that U takes the domain of B ⊂ M Σ into the domain of M Σ . This claim will allow us to do this.
Proof. If g ∈ L 2 Σ then the map U : L 2 Σ → L 2 Σ defined by U g(t) = D(t)g(t) is clearly an isometry since Recall that K A is the space of Cauchy transforms of the measure Σ. The linear map V A is an isometry from K ⊂ L 2 Σ onto K A . We can extend V A to a partial isometry acting on all of L 2 Σ by the formula Then for any f ∈ L 2 Σ , V A f (z) is a column vector with components Since v j (t) = D j1 (t)v 1 (t) + ... + D jn (t)v n (t), it follows that the above can be written as so that On the other hand the Cauchy transform isometry W : Finally, observe that for any g ∈ L 2 Σ , This proves that V A U = W. (9.8) Now U : L 2 Σ → L 2 Σ is an isometry, V A : L 2 Σ → K A is a partial isometry with initial space K and W : L 2 Σ → K A = L( Φ A ) is an onto isometry. If Ran (U ) is not contained in Ker(V A ) ⊥ , then we could find an f ∈ L 2 Σ such that U f = g K + g ⊥ with g K ∈ K and g ⊥ = 0 in L 2 Σ ⊖ K. But then it would follow that which would contradict the fact that We also have that U * f ∈ Dom(M Σ ) so that In summary we have established the necessity half of: with characteristic functions Θ 1 and Θ 2 (where we fix a choice of Θ 2 to obey the condition of Remark (9.2)). If Θ 1 is inner then B 1 B 2 if and only if the following three conditions hold: (1) There exists a contractive C m×m −valued analytic function Φ such that Φ ≥ Θ 1 .
(2) The Herglotz measure Σ of Φ is absolutely continuous with respect to the C n×n −valued Herglotz measure Σ of Θ 2 , for a C m×n matrix-valued function D(t) whose columns divided by t + i belong to L 2 Σ .
where W : L 2 Σ → K A is the deBranges isometry, then for any f ∈ Dom( B 1 ) we have that Proof. To prove the sufficiency half of the above theorem, suppose that the above three conditions are satisfied and choose A ∈ Ext(B 1 ) so that Φ A = Φ (such an A exists by Theorem 8.14).
We know that B 1 is unitarily equivalent to a restriction of M Σ . Here are the details: Let Σ be the matrix-valued measure which is π times the Herglotz measure for Φ A . Let W be the Cauchy transform isometry which takes L 2 Σ onto K A = L( Φ A ) where Φ A is the contractive analytic function corresponding to Σ. Then it is clear that Let U act as multiplication by D(t). The second condition in the above theorem ensures that U : L 2 Σ → L 2 Σ is an isometry. The third condition in the above theorem ensures that this isometry U : L 2 Σ → L 2 Σ maps Dom( B 1 ) into Dom(M Σ ) and since U acts as multiplication by D(t), U B 1 ⊂ M Σ U . In conclusion, This proves the sufficiency of the above three conditions when Θ 1 is inner.
Remark 9.6. The technical assumption on the characteristic function Θ 2 from Remark 9.2 can be easily removed to obtain a fully general result: Consider the Herglotz integral representation of the Herglotz function G Θ2 : By Theorem 8.15, we see that there is a canonical self-adjoint extension Z Θ2 (1) of Z Θ such that Φ[Z Θ2 (1); Z Θ2 ] = Θ 2 , and it follows that P = χ {1} (b(Z Θ2 (1))). In particular if 1 is not an eigenvalue of the Cayley transform U = b(Z Θ2 (1)) of b(Z Θ2 (1)), then it follows from Section 4 that the deBranges Cauchy transform isometry W Θ2 : L 2 Σ2 → L(Θ 2 ) is onto. In this case, as in [4, Section 3.5, Section 5.4], one can check that W Θ2 implements a unitary equivalence between Z Θ2 and M Σ2 , the symmetric operator of multiplication by t in L 2 Σ2 on the domain and moreover that W Θ2 implements a unitary equivalence between Z Θ2 (1), and M Σ2 , the self-adjoint operator of multiplication by t in L 2 Σ2 . By [2, Proposition 5.2.2], it follows that the canonical unitary extension b(B(U )) for U ∈ U(n) has 1 as an eigenvalue if and only if where z ∈ D approaches 1 non-tangentially.
Note that in particular if B 2 is densely defined, then every canonical self-adjoint extension of B 2 is densely defined, and this happens if and only if no unitary extension of b(B 2 ) has 1 as an eigenvalue, so that in this case P = 0, and W Θ2 is onto. More generally the Livsic characteristic function Θ 2 of B 2 is really only defined up to conjugation by fixed unitary matrices. It follows that we can always fix a choice of Θ 2 so that Z Θ2 (1) does not have 1 as an eigenvalue, so that W Θ2 : L 2 Σ2 → L(Θ 2 ) is an onto isometry, and we can assume without loss of generality that B = M Σ2 . That is we fix a choice of Θ 2 so that Ker( lim Alternatively, and perhaps more satisfactorily, it should be possible to remove the technical assumption from Remark 9.2 completely by re-expressing the conditions of the above Theorem in terms of spaces of square integrable functions on the unit circle, and the reproducing kernel Hilbert space on C \ T obtained by taking the Cauchy transforms of such spaces. However as we have preferred to express our results in terms of L 2 spaces on the real line and Herglotz spaces on C \ R, we will not develop the necessary machinery to pursue this here. This is a unitary matrix, and moreover U | Ker(W ) ⊥ = W | Ker(W ) ⊥ , so that U is a unitary extension of W . Note however, that as shown in Example 8.10 that 1 is an eigenvalue of this choice of U . Hence in order to apply Theorem 9.5 we will instead work with a different canonical unitary extension of W . Let Finally as before Σ B = π 1 3 δ −1 + π(1 + β 2 ) 5 12 δ β + π(1 + β −2 ) 5 12 δ β −1 .
It is a bit more tedious to calculate the roots of this polynomial this time. However it is not hard to check that p(i) = 0, and one can verify that p has a double root at z = i and that the third root of p is located at the point µ = i−4 i+4 ∈ C + . It follows that up to a unimodular constant, which is indeed greater or equal to Θ B .
We now show that the Herglotz measure of Φ[A; B](z), is absolutely continuous with respect to the Herglotz measure of Θ T = Φ[A; T ] so that the second condition of Theorem 9.5 is also satisfied: Let us calculate the Herglotz measure Σ T of Θ T = Φ[A; T ]. Now A = b −1 (X), and we calculate σ X , the Herglotz measure of θ X := Θ T • b −1 , as in Example 8.10 by calculating the spectral measure of the unitary matrix X. The determinant of (z − X) can be calculated to be det(z − X) = (z − i)(z − λ)(z + λ) =: p(z), λ := 2 5 √ 6 − i 1 5 .

Outlook
There are several directions in which the results of this paper can be extended.
We have assumed throughout that B ∈ S has an inner Livsic characteristic function. A good portion of the theory we have developed here does not depend on this fact, and it would be good to generalize the results contained here to the case where the Livsic function is an arbitrary contractive analytic function (vanishing at z = i). We have done some work on this already, in particular Example 4.6 can be generalized to show that if Θ ≤ Φ are arbitrary contractive analytic functions that there is a bounded multiplier V : L(Θ) → L(Φ) which intertwines Z Θ and Z Φ . However it is not clear whether Z Θ Z Φ in this general case, or whether more general definitions of partial order, and extensions of a symmetric linear transformation are needed. Also if A ∈ Ext(B) where Θ B is not inner, then one can show that in general H A is only boundedly contained in K A , and so is not just a Hilbert subspace. Once these results are successfully generalized to arbitrary simple symmetric and isometric linear transformations with equal indices, a natural question is whether our partial order results can be extended to arbitrary contractions. Namely given contractions T 1 , T 2 , perhaps one could define that T 1 T 2 if T 1 ≃ T ′ 1 ⊆ T 2 . Perhaps this could be accomplished by using the fact that the problem of unitary equivalence of contractions is equivalent to the problem of unitary equivalence of partial isometries, see [6, Theorem 1] and the discussion following it.
There should be several interesting consequences of the results already obtained in this paper. For example as discussed in Remark 7.4, we can use the theory developed here to provide an alternate proof of the Alexandrov isometric measure theorem, [17,Theorem 2]. In fact the result we obtain is a generalization of the operator theoretic result of Krein [5, Chapter 1, Corollary 2.1] which uses the theory of entire symmetric operators and hence holds for the case where Θ B is a meromorphic scalar-valued inner function. We point out that this result of Krein can be used to prove the Alexandrov isometric measure theorem, and that de Branges has also proven this result in the case where Θ is meromorphic in his book [12,Theorem 32]. Our generalization holds for arbitrary inner functions, and it should be possible to extend this to vector-valued Hardy spaces and matrix-valued inner functions as well. Our theory should also allow us to extend the main result of [23] to the case of arbitrary inner functions and nearly invariant subspaces, as well as to vector-valued versions of nearly invariant subspaces.
Finally as discussed in Remark 8.6, there is a natural bijection between the sets Ext(B) and POVM(B), the set of all unital positive operator valued measures which diagonalize B. It is easy to see with an application of Naimark's dilation theorem that POVM(B) is a convex set, and we think it could be interesting to study the properties of this convex set, for example to determine its extreme points, and to study its Choquet theory. It is known that POVM(B) is a face in the set of all unital positive-operator valued measures on R [24,Theorem 13.6.3], and consequently that every projection valued measure corresponding to a canonical A ∈ Ext(B) is an extreme point of this set (although this can be proven directly). Naimark has proven that if B ∈ S n (H) and A ∈ Ext(B) is self-adjoint in K where K ⊖ H is finite dimensional, then the positive operator-valued measure corresponding to A is an extreme point of POVM(B) [25]. Moreover Gilbert has proven that if B ∈ S n (H), then the set of all Q ∈ POVM(B) which correspond to A ∈ Ext(B) defined on K with K ⊖ H finite dimensional is dense in a natural topology on POVM(B) [26]. It could be interesting to see whether the extreme points of POVM(B) can be given a function theoretic characterization in terms of the characteristic functions Φ[A; B] of the corresponding extensions of B.