Minimal Realizations of Atomic Density Functions

In this article, it is shown that Koebe inner functions and squares of singular Nevanlinna functions, so called atomic density functions, have minimal realizations in reproducing kernel Krein spaces.


Introduction
A real complex function is a meromorphic function of bounded type on the upper half-plane C + which has real boundary values almost everywhere. Here, of bounded type means that the function can be expressed as the quotient of two bounded analytic functions. Real complex functions have recently attracted substantial interest, for example, they were used in [12] to define a continuous analog of the Riesz projection on H p for 0 < p < 1. An important subclass is constituted by density functions, which have positive and not merely real boundary values. It was shown in [14] that every density function h can be factorized into "atomic" parts. More precisely, we can express h as Communicated by Elena Cherkaev.
where V and W are inner functions and q n is a singular Herglotz-Nevanlinna function for all n ∈ Z. Here, an analytic function on C + is a Herglotz-Nevanlinna function if it maps C + into C + and it is singular if it has real boundary values almost everywhere. In this factorization, the first and the second factor carrying the poles and the zeros respectively are called Koebe inner functions. For details on the convergence of the infinite product see [14,Sect. 6]. The methods that have been used in the literature so far have been of function theoretic and classical operator theoretic nature. In this article, we go the first step in constructing a framework to investigate density functions with methods from modern functional analysis. More precisely, our objective is to construct minimal realizations for atomic density functions on Krein spaces, which are indefinite inner product spaces.
A realization of a locally analytic function is essentially a way of expressing the function in terms of the resolvent of a self-adjoint operator. This establishes a link between the properties of the function and the operator. In this case, we can use methods from operator theory to analyse the function or conversely methods from complex analysis to derive properties of the corresponding operator.
As a simple example of a realization, consider the function f (ζ ) := (A − ζ ) −1 u, u H , where A is a self-adjoint operator on a Hilbert space H, u ∈ H and ζ ∈ (A). We also assume that this realization is minimal, i.e. that The function f is analytic in the resolvent set of A and f (C + ) ⊂ C + . Conversely, it can be shown that an analytic function g satisfying g(C + ) ⊂ C + and the growth condition lim y→∞ y|g(iy)| < ∞ is of this form. We point out the following: Since eigenvalues of self-adjoint operators on Hilbert spaces cannot have algebraic multiplicity larger than one and the realization is minimal, it follows that such functions can only have simple poles on the real line. The minimality assumption is essential for this reasoning, which is often the case for using the connection between operator theory and complex analysis.
Up until recently, the theory of realization was mostly restricted to Generalized Nevanlinna functions, which were introduced by Krein and Langer in the 1970s [16]. A Generalized Nevanlinna function is a function of the form q ·h, where q is a Herglotz-Nevanlinna function and h is a rational density function, see [5] or [9]. These functions, their realizations and their connections to other topics such as Weyl functions and the extension theory of symmetric operators have been extensively studied, see e.g. [4] and [6]. However, functions of this type are rather restrictive in terms of possible infinite behaviour, for example they can only have finitely many poles and zeros on C + .
Recently, the theory of realizations has been extended significantly. In [17] it was shown that every meromorphic function of bounded type has a realization, which captures the entire domain on the upper half-plane. However, this realization is not minimal, and it is very likely that this function class is too large to expect minimality in general.
Consequently, an interesting object is the Extended Nevanlinna class, which was originally introduced in the context of system theory and electrical engineering [5].
This class consists of functions of the type q ·h, where q is again a Herglotz-Nevanlinna function and h is now an arbitrary density function. So on the one hand, this class is the natural extension of the Generalized Nevanlinna functions for which existence and minimality holds. And on the other hand, it is a subclass of the functions of bounded type for which existence is established but minimality is questionable. The main result of this article, which is the construction of minimal realizations for atomic density functions, can be seen as a first step in building a minimal realization theory for this extended class.
This article is organized as follows: In the next section, we formally introduce realizations and the closely connected notion of a reproducing kernel Krein space. We then state our main theorem in Sect. 3. In Sect. 4, we review singular Herglotz-Nevanlinna functions and their properties. In Sect. 5, we construct a minimal realization for squares of singular Herglotz-Nevanlinna functions. This result is then transferred to the class of Koebe inner functions using Möbius transformations in the last section.

Realizations
In this section, we formally introduce realizations. To this end, recall that a complex vector space K with an indefinite inner product [·, ·] K is called Krein space if K is decomposable into a direct orthogonal sum where (K + , [·, ·] K ) and (K − , −[·, ·] K ) are Hilbert spaces. We call K a Pontryagin space in the case that either K + or K − is finite dimensional. The set of bounded linear operators between two Krein spaces K 1 and K 2 is denoted by B(K 1 , K 2 ) and we set B(K, K) := B(K). Finally, for a linear relation A on K the adjoint is denoted by A + , for details we refer to [8].
Definition 2.1 Let q : D ⊂ C → C be an analytic function. Then q admits a realization if there exists a Krein space K, a self-adjoint relation A on K with (A) = ∅, a point ζ 0 ∈ (A) ∩ D and an element v ∈ K such that it holds In this case, the pair (A, v) is called a realization of q. Moreover, the defect function φ is defined as Note that φ(ζ 0 ) = v. We sometimes refer to the pair (A, φ) as a realization of q. Finally, the realization (A, v) is called minimal if span{φ(ζ ) : ζ ∈ (A)} = K.

Remark 2.2
The resolvent set (A) is symmetric with respect to the real line since A is self-adjoint. Moreover, if we use the right hand side of (2) to define a function q on (A), it then satisfies q(ζ ) = q(ζ ). Consequently, realizations are usually used for functions that either satisfy this symmetry property or are defined in a subset of the upper half-plane.

Remark 2.3 A calculation involving the resolvent identity shows that
Moreover, the defect function φ is an analytic (vector-valued) function since the resolvent is analytic.
such that for all w ∈ Z : 1. The function ζ → H(ζ, w) belongs to K.

Remark 2.5
The function H is unique, and the kernel functions H(·, w), w ∈ Z form a total set, see [15,Sect. 3].

The Main Result
In this section, we formally introduce density and atomic density functions, and then state our main theorem. To this end, recall that a bounded analytic function V is called inner function if |V (z)| ≤ 1 for all z ∈ C + and |V (z)| = 1 for a.e. z ∈ R.
We call f a density function if it is has positive boundary values almost everywhere. We call f an atomic density function if it is of one of the following forms where V is an inner function and q a singular Herglotz-Nevanlinna function. Functions of the first two forms are called Koebe inner functions.
Recall that the notion of atomic density functions is motivated by the factorization (1).
One reason why density functions, or more generally real complex functions, fit in well in the theory of realizations, is that in this case the extension given by the realization (see Remark 2.2) to the lower half plane is completely natural. In order to specify what we mean by natural, we introduce the classical notion of a pseudocontinuation [11].  We extend a function h with real boundary values by its (unique) pseudocontinuation to the lower half plane, and then analytically continue it through the real line whenever this is possible. We reserve the symbol D ⊂ C + for the original domain and D ext (h) ⊂ C for the extended domain, which is open.
The main result of this article is the construction of minimal realizations on reproducing kernel Krein spaces for atomic density functions. We first define the kernel which is of interest for us: be a meromorphic function of bounded type with real boundary values. We define the Nevanlinna kernel as where we analytically continue if ζ = w.
We point out that the Nevanlinna kernel naturally shows up in the theory of realizations (recall (3)).

The pair (A, φ) defined as
More precisely, the operator R w is defined as Moreover, this realization is weakly unique.
First of all, we clarify what we mean by weakly unique: be two minimal realizations on Krein spaces K 1 and K 2 respectively, which admit ( They are weakly unique if there exists a densely defined and isometric operator with dense range . Note that U does not have to be bounded. This is the natural notion for realizations on Krein spaces, see for example [1]. In summary, the first and second condition in Theorem 3.5 specify the space and the form of the realization respectively, while the third condition means that this realization acts in a natural way on L(h). Realizations of this form are called canonical, following e.g. [10].

Singular Herglotz-Nevanlinna Functions
In this section, we briefly review (singular) Herglotz-Nevanlinna functions. One important property is the existence of an integral representation, for details see [2] and [3]: The following Lemma collects properties of singular Herglotz-Nevanlinna functions. Proof The derivative of the representing measure σ (with respect to the Lebesgue measure) in (4) is given by see [18,Theorem 3.27]. Therefore, the statement (a) is equivalent to D(σ )(ζ ) = 0 for almost every ζ ∈ R. This in turn is equivalent to σ being singular with respect to the Lebesque measure since the absolutely continuous part σ ac of σ is given by [18,Theorem A.45] For the equivalence of the statements (a) and (c) consider the Cayley transform An elementary calculation shows that C defines a one-to-one correspondence between the set of singular Herglotz-Nevanlinna functions and the set of inner functions without the constant inner function V (ζ ) = −1. This means that a singular Herglotz-Nevanlinna function q has a reproducing kernel space L(q) with associated domain D ext (q) and kernel N q . We describe the structure of the space L(q) now. More precisely, we prove that L(q) is isometrically isomorphic to the model space H (V ), where V is the associated inner function of q. This is helpful since model spaces have been studied extensively in the past, see e.g. [13] for a survey.

Proposition 4.4 Let V be an inner function and define its model space H
The proof can be found in [13,Sect. 5].

Proposition 4.5 Let q be a singular Nevanlinna function and V the associated inner function. Then
Proof If the inner function V is constant then both L(q) and H (V ) are trivial. Therefore, we assume without loss of generality that V is not constant. We extend V by its meromorphic pseudocontinuatioñ Note that |V (z)| < 1 because V is not constant. Then for every w ∈ B is a well-defined and analytic function on B (by extending it analytically for ζ = w).
It is a standard fact [13, Proposition 7.14] that for all points w, x ∈ B it holds that Consider the operator This operator is well-defined because for all w ∈ B T (N q (·, w)) = ((V (·) + 1)N q (·, w)) |C + =s V (·, w) |C + · 1 In addition, dom(T ) is dense in L(q) since and consequently B is dense in D ext (q). Recall that N q (·, w) depends continuously on w in L(q). Finally, it is easy to see from (6) and (5) that T is indeed an isometry, and therefore extends to an isometric isomorphism.

Canonical Realizations for q 2
In this section, we construct a canonical realization for the square of a singular Herglotz-Nevanlinna function q 2 out of the canonical realization for q (recall Theorem 4.3). To this end, we first prove two preliminary, but important Lemmas.

Lemma 5.1 Let q be a singular Herglotz-Nevanlinna function. Consider the sets L(q) and q ·L(q) as subsets of O(D ext (q)), which is the set of analytic functions on D ext (q). Then it holds that
Proof Let V be the associated inner function of q. Assume that f + q · h = 0 with f , h ∈ L(q). Then it holds that By Proposition 4.5, it is sufficient to show that both are trivial. It is a standard property of model spaces [13, Lemma 5.10] that they decompose orthogonally as We conclude that Using (7), we see that

Furthermore, since V acts as an isometry on H (V 2 ) and H (V ) and V H(V ) are orthogonal, it follows that
In summary,f + Vf and i(h − Vh) are orthogonal in H (V 2 ) and (8) holds. Therefore, we conclude thatf + Vf = 0. Sincef and Vf are also orthogonal in H (V 2 ), it follows thatf = 0. Finally, this implies that f = 0.

Lemma 5.2 Let q be a singular Nevanlinna function. Then q is analytic at z 0 ∈ R if and only if q 2 is analytic at z 0 .
The proof is postponed to the end of this section. Now we are ready to construct the reproducing kernel Krein space for q 2 . To this end, we equip the set q · L(q) with the inner product which turns q · L(q) into a Hilbert space. Since L(q) ∩q · L(q) = {0}, we can consider the direct product of Hilbert spaces L(q) ⊕ qL(q) as a space of analytic functions instead of pairs of analytic functions.

Moreover, we equip K with the inner product
Then K is a reproducing Kernel Krein space for q 2 with associated domain D ext (q) = D ext (q 2 ).
Proof First of all, the space K is a Krein space of analytic functions on D ext (q) = D ext (q 2 ) (Lemma 5.2). Next, using the identity we see that the kernel functions N q 2 (·, w), w ∈ D ext (q 2 ), are elements of K.
Finally, the kernel functions are also reproducing. To show this, let w ∈ D ext (q 2 ) be arbitrary and f = f 1 + q · f 2 ∈ K, where f 1 , f 2 ∈ L(q). We calculate This finishes the proof.

Remark 5.4
From the definition of the operator , it is easy to see that L(q) and q · L(q) are skewly-linked as subspaces of K. Furthermore, note that is self-adjoint and = I d, which means that is a fundamental symmetry.
Finally, we are able to construct a canonical realization for q 2 , which proves the first part of our main Theorem.

Theorem 5.5 Let q be a singular Nevanlinna function. Then q 2 admits a canonical realization.
Proof We have seen in Theorem 5.3 that q 2 has a reproducing kernel Krein space with associated domain D ext (q) = D ext (q 2 ). We denote it by L(q 2 ). Below, we show that the difference quotient operator is well defined and bounded on L(q 2 ). Let w ∈ D ext (q 2 ) be fixed. In addition, letR w be the difference quotient operator on L(q), which is well defined and bounded by Theorem 4.3. Finally, consider the operator N q (·, w).
Note that G w is bounded and that G * w = G w . Consider the operator T w on L(q 2 ) = L(q) ⊕ q · L(q) given by It is clear that T w is bounded since all operators occurring in the definition are bounded.
We check that T w is the difference quotient operator. By linearity, it is sufficient to verify this for functions f ∈ L(q) and q · f ∈ q · L(q). For f ∈ L(q) this is clear by construction. We calculate for ζ = w Therefore, T w is the difference quotient operator, and we are going to write R w instead of T w subsequently. Next, we define a linear relation A w by

It follows from a straightforward algebraic calculation that
The relation A is also self-adjoint. In order to show this, we first establish using R * w =R w and the fundamental symmetry that Consequently, A is self-adjoint since Finally, we show that A gives rise to a realization. To this end, pick any base point ζ 0 ∈ D ext (q 2 ) and define v = N q 2 (·, ζ 0 ). An elementary calculation shows that Using this and the reproducing property, we arrive at Therefore, the pair (A, v) is a realization. From (11) we also obtain that Consequently, the range of φ forms a total set in L(q 2 ) because it consists of the kernel functions N q 2 (·, w), w ∈ D ext (q 2 ) (recall Remark 2.5). This means that the realization is minimal.

Proof of Lemma 5.2
The "only if" direction is clear. Conversely, let q 2 be analytic at z 0 . Then there is an > 0 such that Assume that z 0 / ∈ D ext (q) and let σ be the measure of the integral representation of q (recall (4)). We know from the theory of Herglotz-Nevanlinna functions that D ext (q) = C \ supp(σ ), where supp(σ ) is the topological support of σ [18,Sect. 3.4]. Consequently, it follows that z 0 ∈ supp(σ ), i.e.
In this section, we prove the main theorem and discuss two illustrative examples. To this end, consider the identity where q is a singular Nevanlinna function and V the associate inner function. This identity allows us to construct a minimal realization for V (V +1) 2 out of the minimal realization for q 2 . In order to treat the inverse function, we use the following theorem, which is a special case of [17, Corollary 3.8]:

a density function which is not identically zero and Z(h) its zero set. Moreover, let (A, v) be a minimal realization of h on a Krein space K with D ext (h) = (A). Then
has a minimal realization (Â,v) on the Krein space (K, [·, ·]K) given bỹ Moreover, it holds that D ext (h) \ Z(h) ⊂ (Â).
The following Lemma can be used to slightly improve the last Theorem. It was proven in [7, Theorem 1.1] for minimal realizations on Pontryagin spaces. However, that proof also works in our more general setting.

Proof of Lemma 5.2 By (3), it holds for any
We apply the functional [·, φ(v)] to both sides with some element x = v ∈ (A) and use (3) to infer that Assume that ζ 0 ∈ σ (A). Let 0 < δ < and v, w ∈ (A) \ B δ (ζ 0 ) be arbitrary. We integrate both sides with respect to x along the circle S 0.5δ (ζ 0 ). The right hand side is analytic on B δ (ζ 0 ). Consequently, its integral vanishes by Cauchys integral theorem. This implies that where P ζ 0 is the Riesz projection onto the spectral set {ζ 0 }. By letting δ → 0 we infer that (14) holds for all v, w ∈ (A). Since the realization is minimal, we know that φ(ζ ) with ζ ∈ (A) is a total set. Consequently, it follows from (14) that P ζ 0 = 0 which contradicts that ζ 0 ∈ σ (A).
Proof of Lemma 5. 2 We only have to verify that (Â) has the right form. From basic function theory, it follows that where Z(h) and P(h) denote the zeros and poles of h respectively. We know that Finally, we can apply Lemma 6.2 to every w ∈ P(h) which yields w ∈ (Â).
Finally, we prove an abstract result linking minimal realizations in arbitrary spaces to canonical realizations. The idea for the isomorphism which is used in the proof comes from [15,Sect. 3].

Proof of Lemma 5.2 Let O(D ext (h)) denote the space of analytic functions on D ext (h).
Consider the operator Then F is well-defined, see Remark 2.3. Moreover, F is also injective since {φ(w) : w ∈ D ext (h)} forms a total set in K by minimality. We equip ran(F) with the indefinite inner product making F an isometric isomorphism of Krein spaces. Thus, ran(F) is a Krein space of analytic functions defined on D ext (h). From (3) it follows that Consequently, N h (·, w) ∈ ran(F). Moreover, the kernel functions N h (·, w), w ∈ D ext (h) are also reproducing since Let z ∈ D ext (h) and w = z ∈ D ext (h). Since F(φ(w)) = N h (·, w), it follows from (3) that This carries over to arbitrary functions f ∈ L(h) because the kernel functions still form a total set since we have only omitted one point.

Proof of Theorem 3.5
In the following, we construct canonical realizations for the functions where V is an inner function. As the first step of this construction, we show that each of these functions admits a minimal realization. To this end, let V be an inner function and consider the associated singular Nevanlinna function q. Then q 2 has a canonical realization (A, v) with (A) = D ext (q 2 ) by Theorem 5.5. Recall that by equality (13) it holds that Consequently, it follows that the pair (A, v) acting on the re-scaled Krein space is a minimal realization of V (V +1) 2 admitting Note that adding constants only corresponds to a change in the constant term in the realization formula (2).
Next, we use Corollary 6.3 to invert the realization of V (V +1) 2 to obtain a minimal realization of (V +1) 2 V . We denote it by (Â,v) and record that it acts on the Krein spacẽ In summary, we have constructed a minimal realization (A, φ) admitting D ext (h) = (A) for every atomic density function h. In view of Theorem 6.4, we can assume without loss of generality that this realization is a canonical realization.
Finally, let (A 2 , v 2 ) be another minimal realization of h admitting D ext (h) = (A). Again, we can assume without loss of generality that this realization is canonical. But it is easy to see that two canonical realizations are weakly isomorphic. Indeed, let (A, φ) be a canonical realization acting on L(h) and (A 2 , φ 2 ) be another canonical realization acting on a reproducing Kernel Krein spaceL(h) (not necessarily isomorphic to L(h)). Then we define a densely defined isometry with dense range by Consequently, the realizations are weakly isomorphic, which finishes the proof.

Remark 6.5
The uniqueness theory for reproducing kernel Krein space is rather involved. See [15] for a survey on this topic.
This function has a canonical realization since tan 2 (ζ ) has one (note the constant term in (2)). We invert this realization using Corollary 6.3 to obtain a canonical realization (A, v) for cos 2 (ζ ) where (A) = C. The function sin 2 (ζ ) is treated similarly.
Author Contributions I have written the manuscript by myself.
Funding Open access funding provided by Stockholm University.

Conflict of interests
The authors declare no conflict of interests.
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