Spectral duality for a class of unbounded operators
We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical approximation of infinite models with suitable sequences of finite models which in turn allow (relatively) easy computations.
Let be an infinite set and let be a Hilbert space of functions on with inner product . We will be assuming that the Dirac masses , for , are contained in . And we then define an associated operator in given by
Similarly, for every finite subset , we get an operator .
If is an ascending sequence of finite subsets such that , we are interested in the following two problems:
(a) obtaining an approximation formula
(b) establish a computational spectral analysis for the truncated operators in (a).
Key words and phrases:Spectrum, approximation, unbounded operator, reproducing kernel, discrete potentials, analysis on graphs, eigenvalue.
2000 Mathematics Subject Classification:18A30, 31C20, 34L16, 34A45, 37A50, 46E22 , 47A75, 47B39
Research supported in part by a grant from the National Science Foundation DMS-0704191
The purpose of this paper is twofold: first to prove that certain linear operators associated with discrete reproducing kernel-Hilbert spaces exhibit spectral duality. This is motivated by more traditional Green’s function techniques for second order elliptic differential operators. Secondly we explore applications of the duality theorem to discrete Laplace operators in weighted (infinite) graphs. In particular we show (for the discrete case) that the Green’s function may be realized as an infinite matrix with entries counting length of paths of edges in a graph.
There has been a recent increase in the interplay between discrete analysis and various continuous limits. While each topic in its own right has been studied for generations, the interconnections are of a more recent vintage, and they in turn have inspired a multitude of exciting new research trends. The motivations for this are manifold, coming in part from numerical analysis, but also more recently from analysis on fractals, see e.g., [DJ07a, DJ07b, BSU08, Str05, Str06], from stochastic processes, from potential theory, Dirichlet forms [Saw97], and discrete Laplacians on weighted graphs [JP08, DJ08, Fab06]. These topics interact with mathematical physics, see e.g., [JP08, DJ08, Pow76, OP96], and with signal processing [DJ07a, DHPS08, Jor83]. But independently of applications, the same themes have an operator theoretic dimension of interest in its own right, see e.g., [HKLW07, Jør78] ; as well as spectral theory [Jør81]. A common thread for this is the use of positive definite functions and reproducing kernel Hilbert spaces [BCR84, Jor89, Jor90, Par70, PS72].
In a variety of studies, the authors have used special subclasses of reproducing kernel Hilbert spaces (RKHSs), and each case appears in isolation; for example, the authors of [DC08] use RKHSs in a systematic study of Fredholm operators, [FJKO05] in potential theory, [GG05, GTHB05, HCDB07] in physics, [HKK07, HCDB07, GTHB05] in signal processing, [Pre07] in statistics, and [SZ08, Tre08] in harmonic analysis. One aim of the present paper is to unify these approaches.
In this paper we take up two themes, one we call spectral reciprocity, and the other is a computational approximation scheme (sections 5 and 6). Both themes interact with the various related developments covered in the above cited papers.
The paper is organized as follows: In section 2 we introduce the Hilbert spaces which admit spectral duality. Let be an infinite set, and let be a Hilbert space of functions on . The crucial restriction on the pair is that the point masses are assumed to lie in the Hilbert space , (Definition 2.1).
The distinction between the discrete and continuous models is illustrated with examples from the theory of stochastic processes. In section 3 we show that the framework of graph Laplacians is included in the setup. Section 4 offers a way of diagonalizing these operators. The idea is analogous to a method used by Karhunen-Loeve (see e.g., [JS07]), but different in that it creates finite matrix approximations to the operator in a global ambient Hilbert space. In section 5 we study approximation: An ascending system of finite subsets in is chosen with union equal to ; and we then show that the corresponding sequence of finite truncations converges. The last theorem identifies a rigorous Green’s function for graph Laplacians.
Thus there are two interesting and interdisciplinary links to operators in symmetric Hilbert spaces (Definition 2.1). It is via operators in these Hilbert spaces built on infinite discrete spaces.
Iterated function systems (abbreviated IFS, [Hut81]) serve in two ways as a link between analysis on discrete systems on one side and operator theory on the other.
Recall that IFSs generate fractal images arising in numerous applications: For example, some IFS-fractals may be built as limits of iterated backwards trajectories of a dynamical system associated to a fixed endomorphism . The generation of the fractals is via recursive procedures applied to branches of a choice of inverse mappings for . As attractors, we then get limit fractal-sets and fractal measures . So in this way the Hilbert space arises as a limit of Hilbert spaces; starting with a graph and passing to the limit.
On the discrete side, the graph has vertices and edges . The first approach (see e.g., [JP08]) is to model IFSs with infinite vertex sets , and associated Hilbert spaces of functions on . In the second approach (e.g., [KU07]) one starts with an IFS, and then there is an associated graph with vertex set a singleton, but instead with edges made up of an infinite set of self-loops.
2. Hilbert spaces of functions
We show that Hilbert spaces of functions which contain the corresponding point masses induce operators arising as graph Laplacians of weighted graphs.
The general setup in our paper is as follows: An infinite set is given, and we consider Hilbert spaces of functions on . One of the Hilbert spaces will be simply . By this we mean the Hilbert space of all functions such that
If , the inner product will be denoted
Let be the set of all finite subsets . Then the expression in (2.1) is by definition
However because of applications, to be outlined later, for a fixed set , it will be necessary for us to consider other Hilbert spaces of functions on .
Let be a Hilbert space of functions on some set . We say that is symmetric if the Dirac functions are in , where
For practical computations we offer in section 4 a method of finite reduction. As an application we give in Corollary 4.7 a necessary and sufficient condition for a Hilbert space of functions to contain its Dirac delta-functions (Definition 2.1).
We will primarily be interested in the case when the set is countably infinite; see especially section 3 below where we will take to be the set of vertices in a given weighted graph. Because of applications to electrical networks, see [JP08] and the references cited there, every weighted graph comes with an associated Hilbert space . In the applications, will denote a space of functions on the vertices of the graph, representing a voltage distribution; and, if , then will be the energy of the configuration represented by .
The following example is different and applies to continuous models; for example models of stochastic processes.
Let . We will be considering functions on modulo constants. Hence the constant function on will be identified with . If is a function on , the derivative is understood in the sense of distributions. Set
Note that if , then and
is well defined. Moreover, the derivative exists pointwise a.e. on . As distributions, and agree.
On , consider the following family of functions indexed by . Set
Writing in the sense of distributions we arrive at the following formula:
For every ,
Hence , and
is not a symmetric Hilbert space; i.e., if then is not in .
(See Definition 2.1 the general case) The condition that is in for all does not imply that is contained in . So there are symmetric Hilbert spaces which do not contain .
If is given, and is a symmetric Hilbert space, we set
Let the vector space of all functions . Then is a linear operator from into .
and we say that is densely defined if is dense in .
Let all finite linear combinations of , i.e., all finitely supported functions on .
Let and be as in the previous definition. A pair of functions: and is said to be a dual pair if
and if the linear span of is dense in .
A dual pair is said to be symmetric iff
Let be as above, and let , be a dual pair. Let be the operator defined in (2.13), and set all finite linear combinations.
Then and is Hermitian on its domain , i.e.,
We have for ,
Thus so , and therefore .
If , then
So if , then
Since the desired conclusion (2.16) holds. ∎
3. Graph Laplacians
We show that every weighted graph induces a Laplace operator and an energy Hilbert space of functions on the vertices of ; and moreover that this setup is included in that of section 2. This is then used in obtaining solutions to a potential theory problem on .
Let be a set. Let be a subset such that if . For , set
We say that is an edge if ; and the points in are called vertices. Further we shall use the notation
Let be a function such that
Further assume for all .
We will assume that is connected, i.e., for every pair there is a finite subset , depending on and such that , and .
The energy Hilbert space . For functions and on , set
More precisely, we will work with functions on modulo the constants. We say that iff
The graph Laplacian.
Let be a weighted graph. We define the graph Laplacian initially on all functions on as follows
In section 2 we started with a symmetric Hilbert space (Definition 2.1), and we derived an associated family of operators from the Hilbert space setup. In this section, the point of view is reversed: we begin with a graph Laplacian and an associated energy Hilbert space. It turns out that the class of operators in section 2 includes all the graph Laplacians.
The energy Hilbert space associated with a weighted graph is symmetric, i.e., for all , we have . Moreover
Let . Then
It is clear that if and .
We finally prove (3.9). Let , and let . Then
Let be a weighted graph; let be the corresponding graph Laplacian, and let be the energy Hilbert space. Let be a function on the vertices satisfying
(a) finite linear span of );
Then there is a such that
Before proving the theorem, we show by a simple example that neither of the two restrictions (a) or (b) on the function may be dropped. We will give examples when some function does not satisfy one of the two conditions. While there will always be a function which satisfies (3.10), the point is that none of the solutions will be in , i.e., the solutions will have infinite energy, i.e., .
Let . By this we mean that has
Fact 1. The only solutions to the equation
have the form , where and , are constants.
Fact 2. On set
Then , i.e., , and
Combining the two facts, we see immediately that the equation
has no solutions in . Note that , but does not satisfy condition (b) in the theorem.
on does not have any solutions in . Note that the function on the right hand side in (3.17) does satisfy (b), but is not in .
We now turn to the proof of Theorem 3.5. The following lemma is helpful:
Let be a weighted graph, and let denote the linear space of functions satisfying conditions (a)-(b) in the statement of Theorem 3.5. Then
Induction on . ∎
Proof of Theorem 3.5.
By the lemma, it is enough to show that for any pair , , the equation
has a solution .
Now fix and in . Using Riesz’ lemma, we first prove that there is a unique such that
Since is connected, there is a finite subset such that , and . Then
Hence the existence of a solution in (3.19) follows from Riesz’ lemma applied to .
We note that satisfies (3.18). Indeed, for all , we have
Hence the two sides in equation (3.18) agree as functions on , and the proof is complete.
Positive semidefinite. Let be a set. Set
A function is said to be positive semidefinite iff
(a) Let be a function. Then the following conditions are equivalent:
|(3.23)||There is a Hilbert space and a function such that|
(b) We say that two systems , in (a) are unitarily equivalent if there is a unitary isomorphism such that
(c) If and are two systems both satisfying (3.24) then and are unitarily equivalent iff
Let be a weighted graph satisfying the conditions in Theorem 3.5. Let be the energy Hilbert space and the graph Laplacian.
(a) For every let be the unique solution in to equation (3.18). Then for a fixed , the function , is positive semidefinite. Moreover, the function , is positive semidefinite.
(b) Let be as in (a), and let be a function satisfying the conditions in Definition 3.1. Let be
Then is positive semidefinite.
4. Diagonalizing subsystems
It is known that positive semidefinite functions define reproducing kernel Hilbert spaces. In this section we identify which of these Hilbert spaces are symmetric (Definition2.1). And we solve the problem of diagonalizing finite subsystems.
Let be a set, and let be a positive semidefinite function. We will consider solutions to condition (3.24), i.e.,
The next result shows that when restricting to finite subsystems, , finite, we may assume that the set is linearly independent in .
Let be positive semidefinite. Let be the space of all finite linear combinations
the kernel of .
Then , i.e., with ; and
We refer to [Aro50] for the general theory of reproducing kernels.
In the analysis below, the idea is to select finite subsets of a fixed ambient infinite set ; and it is assumed that is a symmetric Hilbert space of functions on . This method of finite reduction is motivated by computations, in that infinite sequences do not admit representations in computer registers.
Since every positive semidefinite function induces a reproducing kernel Hilbert space via Definition 4.1, it is important to note that the class of Hilbert spaces in Definition 2.1 are restricted in two ways: a symmetric Hilbert space is a space of functions on a given set and for all .
The following example shows that may be obtained from a positive semidefinite function , even though is not a space of functions on .
Let and be compactly supported distributions, and the tensor product where the right-hand side is evaluation on , written , . The -inner product is defined by
where the right-hand side now denotes application of the distribution to .
If , , are the distribution derivatives, then
is an orthonormal basis in . Indeed, if for some then , and the expansion in is as follows
and for , we have
i.e., the Taylor expansion.
Because of Lemma 4.2, we will assume in the sequel that when and are as described then is not in .
Let be a positive semidefinite function, and let be the Hilbert space in Definition 4.1. Let be a finite subset, and set
Let be an ONB in satisfying
For , set
Then is an ONB in , and
We first show that the system in (4.12) is orthonormal in . Let . Then
By Lemma 4.2 we see that is indeed an ONB for and that
is the orthogonal projection onto . Note that we use Dirac’s “ket-bra” notation on the right hand side of (4.14).
We now prove (4.13). For we have
which is the desired formula (4.13). ∎
Recall is a symmetric Hilbert space iff for all . Assume this condition holds; and let the set of all finite subsets of , and let .
From (4.14), recall the formula for the projection onto :
Since , we have