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Selfadjoint extensions of the multiplication operator in de Branges spaces as singular rank-one perturbations

We derive a description of the family of canonical selfadjoint extensions of the operator of multiplication in a de Branges space in terms of singular rank-one perturbations using distinguished elements from the set of functions associated with a de Branges space. The scale of rigged Hilbert spaces associated with this construction is also studied from the viewpoint of de Branges's theory.


Introduction
In de Branges's theory of Hilbert spaces of entire functions, the operator of multiplication by the independent variable plays a central role. From particular features of this operator, one can infer properties of concrete realizations of de Branges spaces. Conversely, the particularities of a de Branges space determine the spectral properties of the selfadjoint extensions of the corresponding multiplication operator. This fact is specially useful when, via the so-called functional model, one can identify Krein's entire operators and, more generally, n-entire operators (in particular, regular and singular Schrödinger operators as well as Jacobi operators) with the multiplication operator in certain de Branges spaces [14][15][16][18][19][20][21].
This article uses singular perturbation theory for dealing with the family of canonical selfadjoint extensions S γ of the multiplication operator S in a de Branges space B. Our approach to this issue is not reduced to the application of perturbation theory to a concrete family of operators. Instead, we focus our attention to functions in de Branges space theory that play a central role when considering the operators S γ as a family of singular rank-one perturbations of a certain selfadjoint extension of S. To the best of our knowledge, this way of dealing with the matter is new. We restrict our considerations to the case when the operator S is densely defined since the other case has already been treated in [22]. As a matter of fact, this work can be regarded as a further development of the results given in that paper. Singular perturbations are treated by means of triplets of Hilbert spaces [4,Chapter I]. We combine this operator-theoretic approach with the properties of functions in both the de Branges space and its set of associated functions assoc B (see Section 2). In doing so, we have two goals in mind. The first one consists in shedding light on the properties of the linear spaces involved in the theory of triplets of Hilbert spaces. The second aim concerns the incorporation of de Branges's theory into the theoretical framework of singular rank-one perturbations. The results concerning the first goal can be summarized as follows: Let B (γ) +2 and F +1 be dom(S γ ) and dom(S * ) with their respective graph norms. Let B (γ) −2 and F −1 be their respective duals. By standard theory, one has B (γ) +2 is a de Branges space (Theorem 3.1) and the space F +1 share many of the properties of a de Branges space (Theorem 3.2). We next prove that F −1 is realized by (that is, is isometrically isomorphic to) a de Branges space which is setwise equal to assoc B (Proposition 3.3 and Theorem 3.4); we remark that F −1 is initially a Hilbert space of continuous linear functionals acting on (the Hilbert space of entire functions) F +1 . Finally, we show that B (γ) −2 is not a de Branges space but rather a quotient space involving assoc B (Theorem 3.6).
On the subject of the second goal, our approach allows us to find formulae for rendering the family of selfadjoint extensions of the multiplication operator as a family of rank-one singular perturbations. Namely (Theorems 4.7 and 4.9), where k(z, w) is the reproducing kernel in B and the functions s γ (z) are given in (2). Then the operator S γ is the restriction of to dom(S γ ); we note thatS γ are maps from dom(S * ) to B (π/2) −2 (the details are discussed in Section 4). Although, in an abstract setting, this kind of formulae are known, here they are derived using function theoretical methods pertaining to de Branges theory that make clear the prominent role played by the functions s γ (z) in these formulae. Furthermore, we obtain as by-products the Krein's formula for resolvents and some objects related to it in terms of functions in de Branges's theory.

Remarks on de Branges Hilbert spaces
Throughout this paper, inner products in Hilbert spaces are assumed conjugate linear with respect to the left argument. We follow the customary rule of denoting a function f by its evaluation at an arbitrary value of its argument f (z). Also, we often denote the action of an operator B on a function f (z) by Bf (z) instead of (Bf )(z).
A Hilbert space of entire functions B is a de Branges (dB) space if it has a reproducing kernel and is isometrically invariant under both the conjugation f (z) → f # (z) := f (z) and the mapping whenever w ∈ C is a non real zero of f (z). Alternatively, dB spaces can be defined in terms of functions of the Hermite-Biehler class HB, that is, entire functions for which the inequality |e(z)| > |e(z)| holds for all z ∈ C + . Indeed, given e(z) ∈ HB, one defines where H 2 (C + ) is the Hardy space on the upper half plane; the inner product in B(e) is given by According to [7, Problem 50] B(e) so defined is a dB space with reproducing kernel for every f (z) ∈ B and w ∈ C such that h(w) = 0. The set of associated functions is denoted assoc B. By [13,Lemma 4.5] one has assoc B = zB + B.
Within assoc B lies the family of functions These functions are in bijective correspondence with the family of canonical selfadjoint extension of the multiplication operator; see below. We note that, in general, s γ (z) ∈ assoc B \ B with the possible exception of at most one of such functions [5,Lemma 7]. Extending the definition (2) to γ ∈ R one has the identity s γ (z) = cos(γ − γ 0 )s γ 0 (z) + sin(γ − γ 0 )s γ 0 +π/2 (z), where γ 0 is fixed but otherwise arbitrary. Also, The operator of multiplication by the independent variable is defined by (Sf )(z) = zf (z) with domain dom(S) maximal in B. This operator is closed, completely nonselfadjoint and has deficiency indices (1, 1). Additionally, S is also regular, i. e. its spectral core is empty, if and only if B = B(e) with e(z) ∈ HB devoid of zeros in the real line. Also, dom(S) may be either dense in B or has codimension equal to one; the latter happens if and only if there exists (a necessarily unique) γ ∈ [0, π) such that s γ (z) ∈ B, in which case s γ (z) is orthogonal to dom(S) [7].
From this point on, we consider only dB spaces with the property of S being densely defined; dB spaces with dom(S) having non zero codimension have been discussed in [22].
Thus, since we assume S densely defined, S * can be described as The canonical selfadjoint extensions of S are the selfadjoint restrictions of S * . A standard description of them is given in [13], where the connection to the family of functions s γ (z) is made explicit: It follows from (6) that spec(S γ ) = {zeros of s γ (z)} and the corresponding eigenfunctions are of the form If S is also regular, then γ spec(S γ ) = ∅, γ spec(S γ ) = R, and spec(S γ ) and spec(S γ ′ ) interlace whenever γ = γ ′ . Then yielding the desired result after one more use of (7).
Another consequence of (7) is where is the Cayley transform. This in turn yields another characterization of the adjoint operator.
Lemma 2.2. Given γ ∈ [0, π) and w ∈ C : im(w) > 0, the adjoint operator S * can be described as follows: With the help of (8), this can be written as belongs to the domain of S γ (see (9)). Since, U(w) + I = 2(S γ − (re w)I), which can be verified by a straightforward computation, (10a) has been established.
To prove (10b), it suffices to note that and rewrite wU(w) + wI using (9).
The description (10) of S * takes its simplest form when w = i (or w = −i). In such a case, one obtains (cf. [3]) Remark 1. From this point on, de Branges spaces are assumed to be generated by Hermite-Biehler functions free of zeros on the real line. Equivalently, de Branges spaces are supposed to have the property that, for every x ∈ R, there exists f (z) in the space such that f (x) = 0. This assumption, which means that the operator of multiplication is regular, entails no essential loss of generality.

Scales of dB spaces
Most of the following discussion is based on [1][2][3] with some slight modifications.
Given γ ∈ [0, π), let B (γ) +2 := dom(S γ ) equipped with the graph inner product Due to S γ being closed, B +2 is a Hilbert space. Its dual B (γ) −2 is the completion of B under the norm induced by the inner product Since the operator S γ has a unique contractive continuationŜ γ : The latter operators satisfy the resolvent identityR for w ∈ C \ spec(S γ ) and v ∈ C \ spec(S γ ). In terms ofR γ (w), the associated pairing ·, · 2 : B (γ) +2 → C can be written as Before going into our first result, let us note that when g(z) and f (z) are related as in (6a).
+2 is a dB space.

Proof. The linear functional of point evaluation is continuous in
Let k(w, z) be the reproducing kernel in B. Then, which along with (14) imply the assertion. Moreover, it follows that so point evaluation is uniformly continuous over compact subsets of C.
Next, let us verify that the conjugation # is an isometry in B proving the assertion; the fact that the Cayley transform is unitary has been used along the way. Now, suppose that v 0 is a non-real zero of g(z) ∈ B (γ) +2 and consider f (z) ∈ B such that Thus, the assumption g(v 0 ) = 0 yields where again the isometric character of the Cayley transform has been used. The proof is now complete.
Let us introduce a second scale of spaces associated with B and S. Define F +1 := dom(S * ) equipped with the graph inner product f, g +F := S * f, S * g + f, g .
Due to (11a), for f (z) and g(z) in dom(S * ), one has From these expansions, using (11b), one obtains This clearly implies f +F = f +2 whenever f ∈ dom(S γ ), which gives rise to the scale B +2 is not dense in F +1 , a fact obvious upon inspection of (15). For the next result we will use the following consequence of Lemma 2.1, where U(w) is the Cayley transform (9). Proof. Given f (z), g(z) ∈ dom(S * ) and w ∈ C, which implies that point evaluation is continuous in F +1 .
Denote by ·, · F the pairing between F −1 and F +1 . The expression f, φ F for f ∈ F + and φ ∈ F −1 of course means φ, f F . Proposition 3.3. The mapping k(·, z), · F is a bijection between F −1 and assoc B.
Given w ∈ C, choose γ such that w ∈ spec(S γ ). For any g(z) ∈ B, where n(z, w) := g(w)R γ (w)ψ(z) − f (w)R γ (w)g(z); Lemma 2.1 has been used in the computation above. Since n(z, w) ∈ B (as a function of z), it follows that f (z) ∈ assoc B. Therefore, k(·, z), · F maps F −1 into assoc B. Suppose now f (z) ∈ assoc B, that is, f (z) = zg(z)+h(z) for some g(z), h(z) ∈ B. Consider {g n (z)} n∈N ⊂ dom(S) such that g − g n → 0 and define f n (z) := zg n (z) + h(z). Since for any l(z) ∈ dom(S * ), it follows that {f n (z)} n∈N is Cauchy convergent in F −1 . Let φ ∈ F −1 be its limit, which is well defined (i. e., independent of the sequence) by standard arguments. Since k(z, w) ∈ dom(S * ) for all w ∈ C, we can definẽ f (z) := k(·, z), φ F . Now, on one hand we have On the other hand, we have assoc B = B(e 1 ) where we can choose e 1 (z) = (z + i)e(z). Let k 1 (z, w) be the reproducing kernel in B(e 1 ) [17]. Then, Therefore, can be made arbitrarily small in compact subsets of C due to (18) and (19). This in fact shows that k(·, z), · F : F −1 → assoc B is onto. Since k(·, z), φ F ≡ 0, one has φ = 0. Thus this mapping is also injective.
On assoc B -and with some abuse of notation-define the norm with f (z) and ψ related as in (17). This makes k(·, z), · F : F −1 → assoc B an isometry and, taking into account the underlying inner product, assoc B is a Hilbert space.
Theorem 3.4. The Hilbert space assoc B given above, i. e. with the norm (20), is a dB space.
In view of Proposition 3.3 and Theorem 3.4, we henceforth identify F −1 with its realization as the Hilbert space assoc B which is equipped with the norm (20). Following up this identification, we define f, g F := ψ, g F , f (z) ∈ assoc B, g(z) ∈ dom(S * ), where ψ ∈ F −1 is related to f (z) by (17). In particular, The next result provides an explicit formula for the duality pairing between F +1 and F −1 .
Proof. We have F −1 = B(e F ) for some e F (z) ∈ HB. Also, since B = B(e) for some e(z) ∈ HB free of real zeros (see Remark 1), F −1 = B(e 1 ) setwise with e 1 (z) := (z + i)e(z). Thus, if I(g) denotes the right hand side of (21), then where the second inequality is implied by the fact that the norms induced by e F and e 1 are equivalent. Now, since it follows that, for some possibly different C > 0, so I defines a continuous functional on F +1 . Finally, it is easy to verify that I coincides with f, · whenever f (z) ∈ B so it is equal to ψ, · F .
Let M (γ) 0 ⊂ F −1 be the annihilator of dom(S γ ), that is, where f (z) ∈ assoc B is any representative of F. Conversely, to every ϕ ∈ B Proof. The trueness of the first part of the statement is rather obvious, since for every element g(z) of dom(S γ ) one has g +2 = g +F . Indeed, for all f (z) within the equivalence class F. As for the second part of the statement, consider ϕ ∈ B (γ) −2 . Given c ∈ C, define ϕ c ∈ F −1 by the rule where h(z) ∈ dom(S γ ) and b ∈ C are related to g(z) ∈ dom(S * ) by the decomposition defining (11a) [3, Lemma 1.3.1]. By Proposition 3.3, ϕ c can be identified with some f c (z) ∈ assoc B. Clearly, Thus, the associated element in assoc B/M

Singular rank-one perturbations
Let us turn to singular rank-one perturbations of selfadjoint extensions of the operator S. To keep the notation simple, let us fix γ = π/2. We show below that the "correct" perturbation of the operator S π/2 is performed along the function s 0 (z) in the sense that S is precisely S π/2 restricted to those f (z) ∈ dom(S π/2 ) that obey s 0 , f 2 = 0. Lemma 4.1. Assume µ ∈ spec(S π/2 ). Then, k(·, µ), g 2 = g(µ) for every g(z) ∈ assoc B.

Lemma 4.2.
For every w ∈ spec(S π/2 ), Proof. We need to show that for every f (z) ∈ B. Due to the continuity of the inner product, it will suffice to show the assertion on elements of a basis of B. Thus, let us consider k(z, µ n ) with µ n ∈ spec(S π/2 ). We have where we have used the previous lemma and the fact that s 0 (z) is real entire. Recalling (4) we have k(w, µ n ) = s π/2 (w)s 0 (µ n ) π(w − µ n ) , whence the assertion follows.
The assertion now follows straightforwardly.
Then, S γ as described by (6) is the restriction ofS γ to dom(S γ ).