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 < +∞, wheref k is the usual Fourier coefficientf k := 1 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 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 Sect. 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 a hypothesis involving the boundedness of operatorvalued 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 Sect. 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 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 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 self-adjoint operator and we denote by H (k) the Hilbert space consisting of 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 be a sequence of abstract trace maps, i.e., τ k is a linear, continuous and surjective map such that its kernel 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 define with corresponding norms · , · • , | · |, | · | • . We denote by ||| · ||| the operator norm of bounded linear operators.

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 K (τ ) is dense in H . For any z ∈ ρ(A k ), let us define the following bounded linear operators: By resolvent identity one has Now let us take z = ±i in the above definitions and pose Then z → k (z) is a Weyl function (equivalently a Krein's Q-function), i.e., it satisfied the identities and k (z) * = k (z).
Therefore, the set is not void: C\R ⊆ Z k (see e.g. [11, Theorem 2.1]). Posing one has the identities is a linear bounded right inverse of τ k . Moreover, since R k :
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 where in this case we used the notation Moreover, 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 :

Then the linear mapτ
is continuous, surjective and its kernel K 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 countable 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 S k := A k |K (τ k ), where A k and τ k are defined as in the previous section.  (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 so |||G (k) ||| = 1.

Remark 3.3 By the definition of D( A , ) one has that A , is a direct sum if and only if both and 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 ( , ) is such that is an orthogonal projection in h = ⊕ k∈Z h k and is a self-adjoint operator in the Hilbert space associated with . The statement of Theorem 3.2 remains unchanged replacing τ k withτ k and G (k) (z) with 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 2 d (N) is continuous, surjective and has a kernel dense in 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 3.7 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 K (z; x, y) = e −z |x−y| − e −z (x+y) 2z , 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 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