Direct sums of trace maps and self-adjoint extensions

We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is a trace map. An application to the theory of self-adjoint extensions of direct sums of symmetric operators is provided; this gives an alternative approach to results recently obtained by Malamud-Neidhardt and Kostenko-Malamud using regularized direct sums of boundary triplets.


Introduction
We begin with a simple example. Let ∆ 0 = ∂ 2 ∂x 2 + ∂ 2 ∂θ 2 be the Laplace-Beltrami operator on the two-dimensional cylinder M 0 := R + × T with respect to the flat Riemannian metric g 0 = ( 1 0 0 1 ). Its minimal realization with domain C ∞ c (M 0 ) is symmetric and negative as a linear operator in the Hilbert space L 2 (M 0 ) = L 2 (R + ) ⊗ L 2 (T). We denote its Friedrichs' self-adjoint extension by ∆ D 0 ; it corresponds to imposing Dirichlet boundary conditions at the boundary T, i.e. D(∆ D 0 ) = {u ∈ H 2 (M 0 ) : lim x↓0 u(x, θ) = 0}. Here H 2 (M 0 ) is the usual Sobolev-Hilbert space of order two. Let us denote by H s (T) the (fractional) Sobolev-Hilbert space of square-integrable functions f on the 1-dimensional torus T such that k∈Z |k| 2s |f k | 2 < +∞, wherê f k is the usual Fourier coefficientf k := 1 √ 2π T e −ikθ f (θ) dθ. Then is a concrete example of what we call an abstract trace map (see the next section), i.e. γ 0 is continuous (w.r.t. graph norm), surjective and its kernel is dense in L 2 (M 0 ). By partial Fourier transform with respect to the angular variable one gets . On D 0 one can define the trace map which is bounded, surjective and with a kernel dense in L 2 (R + ). Moreoverγ 0 is bounded uniformly in k ∈ Z w.r.t. the graph norm of d 2 k , and so the infinite direct sum is a well defined bounded operator. Since γ 0 corresponds to ⊕ k∈Zγ 0 by partial Fourier transform, (1.1) does not define a trace map since it is not surjective: its range space is the strict subspace of ℓ 2 (Z) defined by This simple example shows that an infinite direct sum of trace maps can fail to be a trace map: the direct sum of the range spaces can be different from the range space of the sum.
In Section 2 we provide a simple criterion which selects the right range space in order that the direct sum of trace maps is a trace map. Such a simple criterion uses an hypothesis involving the boundedness of operator-valued sequences obtained composing the trace maps with their right inverses (see (2.1)). Such a hypothesis seems a very strong one (indeed that allows an easy proof), however we show that always there exist right inverses such that (2.1) holds true (see Lemma 2.3).
In Section 3 we give an application to self-adjoint extensions of direct sums of symmetric operators and provide a couple of examples. We obtain that the methods here presented permit to obtain results equivalent to the ones recently obtained in [8] and [7] using regularized boundary triplets (see Remark 3.5).
In Example 1 we determine the trace space for the evaluation map In this case Theorem 2.1 easily implies that the range space is a weighted ℓ 2 -space with weight w n = d −1 n , where d n := x n+1 − x n . By Theorem 3.2 such a trace map can be used to define one-dimensional Schrödingier operators with δ and δ ′ interaction supported on the discrete set X, thus providing a construction alternative to the one presented in [7].
In Example 2 we show that our criterion easily gives the correct trace space H 1 2 (T) for the example provided at the beginning. Then we point out that the same criterion allows to prove that H s (T), s = 1 2 − α 1+α , is (isomorphic to) the defect space of ∆ min α , −1 < α < 1, the minimal realization of the Laplace-Beltrami operator ∆ α : ∂θ 2 corresponding to the degenerate/singular Riemannian metric g α (x, θ) = 1 0 0 x −2α . We refer to the papers [3] and [4] for the almost-Riemannian geometric considerations leading to the study of ∆ α and to [12] for the classification of all self-adjoint extensions of ∆ min α .

Direct sums of abstract trace maps
Let H k , k ∈ Z, be a sequence of Hilbert spaces, with scalar product ·, · k and corresponding norm · k . On each H k we consider a selfadjoint operator A k : D(A k ) ⊂ H k → H k and we denote by H (k) the Hilbert space made by D(A k ) equipped with a scalar product ·, · (k) giving rise to a norm · (k) equivalent to the graph one.
Let h k , k ∈ Z, be a sequence of auxiliary Hilbert spaces with scalar product [·, ·] k and corresponding norm | · | k . Let τ k : H (k) → h k , k ∈ Z , be a sequence of abstract trace maps, i.e. τ k is a linear, continuous and surjective map such that its kernel K (τ k ) is dense in H k . Since τ k is continuous and surjective there exists a linear continuous right inverse (see e.g. [2, Proposition 1, Section 6, Chapter 4]). Since τ k is surjective, ι k is injective and so we can define a new scalar product on h k by Let us denote by h (k) the Hilbert space given by h k equipped with the scalar product [·, ·] (k) . We pose with corresponding norms · , · • , | · |, | · | • .
Theorem 2.1. Let ι k be a linear continuous right inverse of τ k and suppose that Then the linear map is an abstract trace map, i.e. is continuous, surjective and its kernel Remark 2.2. Notice that Theorem 2.1 holds true for any sequence of Hilbert spaces H (k) , k ∈ N, such that each H (k) is densely embedded in H k . However our hypotheses H (k) = D(A k ) permits to show that it is always possible to find right inverses ι k such that hypothesis (2.1) is satisfied (see Lemma 2.3 below).
For any z ∈ ρ(A k ), let us define the following bounded linear operators: Now let us take z = ±i in the above definitions and pose .
. Therefore the set Proof. By (2.5) one has Remark 2.4. In the case there exists λ ∈ ∩ k∈Z ρ(A k ) ∩ R the previous reasonings have the following variant. By (2.2) there follows and so Since it is self-adjoint and its range is closed (since the range of G k (λ) is closed), G k (λ) * G k (λ) is a continuous bijection. Then Moreover, by using the scalar product . and, proceeding as in the proof of lemma 2.3, |||ι k τ k ||| = 1 .
Theorem 2.1 has the following alternative version where one can still use the original trace space h as long as one regularizes the traces τ k : Theorem 2.5. Let us define r k := (G * k G k ) 1/2 and τ k : is continuous, surjective and its kernel K (τ ) = K (τ ) is dense in H .
Proof. The proof is the same as in Theorem 2.1. It suffices to notice thatι k := ι k r k is the right inverse ofτ k and that Remark 2.6. Notice that in this section Z can be replaced by any other denumerable set N and that we can replace [·, ·] (k) by a scalar product inducing an equivalent norm. Moreover, given a finite subset F ⊂ N, we can replace h (k) by h k for any k ∈ F .

Applications and Examples.
Let S k , k ∈ Z, be the sequence of symmetric operators defined by is considered as a map on H (k) to h (k) , so that when calculating the adjoint G (k) (z) of τ k (R k (z)) one gets Next Lemma shows that the direct sums ⊕ k∈Z G (k) (z) appearing in Theorem 3.2 below are well defined bounded operators: Proof. By (2.2) one has, posing G (k) := G (k) (−i), By (2.6),  and Θ is a self-adjoint operator in the Hilbert space Range(Π). Denoting by A Π,Θ the self-adjoint extension associated with (Π, Θ) one has Moreover, for any z ∈ (∩ k∈Z ρ(A k )) ∩ ρ(A Π,Θ ), Here and z • ∈ ∩ k∈Z ρ(A k ).

Remark 3.3. By the definition of D(A Π,Θ ) one has that A Π,Θ is a direct sum if and only if both Π and Θ are direct sums.
In the case z • ∈ R one has G ⋄ (k) = 0 and Remark 3.4. Theorem 3.2 has an alternative version in the case one uses the trace map furnished by Lemma 2.5. In this case the extension parameter (Π, Θ) is such that Π is an orthogonal projection in h = ⊕ k∈Z h k and Θ is a self-adjoint operator in the Hilbert space associated with Π. The statement of Theorem 3.2 remains unchanged replacing τ k withτ k and G (k) (z) with Remark 3.5. By [10] and [11,Section 4], Theorem 3.2 and Lemma 2.5 provide results equivalent to the ones that can be obtained using Boundary Triplet Theory. Let us for simplicity take z • = i. Then (see [10,Theorem 3.1]) is a boundary triple for S * k , i.e. β k,1 and β k,2 are surjective and satisfy the Green-type identity (see [10,Theorem 3.1]). By (3.1) there follows that {h k , r k β k,1 , r −1 k β k,2 }, where r k is defined in Lemma 2.5, is a boundary triple for S * k as well with Weyl function r −1 k M k (z) r −1 k . By Lemma 2.5 and [10, Theorem 1.6] one gets Let us notice that the norm of r −1 k τ k ≡τ k : H (k) → h k is bounded uniformly in k ∈ Z by Lemma 2.5, and soβ 0 is well defined.β 1 is well defined as well by the definition of D(S * ). In For any n ≥ 0, the map τ n is continuous, surjective and has a kernel dense in L 2 (I n ).
By Remark 2.6 we can suppose n = 0 and, since 0 ∈ ∩ n>0 ρ(A n ), we can use the results provided in Remark 2.4 with λ = 0.
The kernel of (−A n ) −1 , n > 0, is given by where θ denotes Heaviside's function and d n := x n+1 − x n . Therefore and so by straightforward calculations one gets that G * n G n : C 2 → C 2 corresponds to the positive-definite matrix G * n G n ≡ d n 1/3 1/6 1/6 1/3 .
In conclusion on h (n) = C 2 we can put the equivalent scalar product Hence, by Theorem 2.1, denoting by ℓ 2 d (N) the weighted ℓ 2 -space one gets that is continuous, surjective and has a kernel dense in By using Theorem 3.2 with trace map τ defined in (3.2), one gets the same kind of self-adjoint extensions given in [7] (the case in which 0 < d * ≤ d * < +∞ has been studied in [6]). Such extensions describe one-dimensional Schrödinger operators in L 2 (−∞, x ∞ ) with δ and δ ′ interactions supported on the discrete set X = {x n } n∈N . These operators have been studied in [1, Chapters III.2 and III.3], when 0 < d * ≤ d * < +∞ and x ∞ = +∞, and in [7] when d * < +∞. Analogous considerations, with A n given by the one-dimensional Dirac operator with Dirichlet boundary conditions on the interval I n , lead to self-adjoint extension describing one-dimensional Dirac operators with δ and δ ′ interactions on the discrete set X = {x n } n∈N (see [1,Appendix J], for the case X in which is a finite set and [5] for the general case).
Example 2. At first let us check that applying Thereom 2.1 to the example given in the introduction one gets the right trace space h • = h 1 2 (Z). Hence here A k = d 2 0 − k 2 . By Remark 2.6 we can suppose k = 0 and, since 0 ∈ ∩ k∈Z\{0} ρ(A k ), we can use the results provided in Remark 2.4 with λ = 0. Since the kernel of (−d 2 0 + z 2 ) −1 , Re(z) > 0, is given by one easily gets and so G * k G k : C → C is given by the multiplication by the real number Therefore h (k) = C is equipped with the scalar product [ξ, ζ] (k) := |k| ξ · ζ and so By using Theorem 3.2 with trace map then one can determine all self-adjoint extensions of the minimal Laplacian on M 0 . Such an example can be generalized in the following way: let M α be R + × T endowed with the singular/degenerate Riemannian metric The Riemannian volume form corresponding to g α is dω = x −α dxdθ and so we denote by L 2 (M α ) be the Hilbert space In [4] it is shown that the minimal realization corresponding to g α is essentially self-adjoint whenever α / ∈ (−3, 1), has deficiency indices (1, 1) whenever α ∈ (−3, −1] and has infinite deficiency indices whenever α ∈ (−1, 1). Therefore, in order to determine and then study all self-adjoint realizations of ∆ min α , −1 < α < 1, by Theorem 3.2 one needs to characterize the range space of the trace map acting on function in the domain of the Friedrichs extensions ∆ D α (corresponding to Dirichlet boundary conditions at T) of ∆ min α (see [12]). Let us sketch here a proof in the case 0 < α < 1, referring to [12] for more details and for the (more involved but still using Theorem 2.1) proof that holds in the case −1 < α < 1.
By partial Fourier transform one gets where L 2 w (R + ) is the weighted L 2 space D α,k := {f ∈ L 2 w (R + ) ∩ C 1 (R + ) : (d 2 α − k 2 q α ) ∈ L 2 w (R + ) , f (0) = 0} . By Remark 2.6 we can suppose k = 0 and, since 0 ∈ ∩ k∈Z\{0} ρ(A k ), A k = d 2 α − k 2 q α , whenever 0 < α < 1, we can use the results provided in Remark 2.4 with λ = 0. Since f ξ ≡ G k ξ solves the boundary value problem Therefore G * k G k : C → C is given by the multliplication by the real number and so h (k) = C is equipped with the scalar product