Operators on hilbert space pdf

Now the variety of applied problems that lend themselves to com- putation of deficiency indices and the study of selfadjoint extensions are vast and diverse. As a result, it helps if one can identify additional structures that throw light on the problem. Here duality considerations within the framework of Hilbert space are tested tools in applied math- ematics.

In this paper we will device such a geometric duality theory: We will attack the problem of computing deficiency spaces for a single Hermitian operator with dense domain in a Hilbert space which occurs in a duality relation with a second Hermitian operator, often in the same Hilbert space. We will further use our duality to prove essential selfadjointness of families of Hermitian operators that arise naturally in reproducing kernel Hilbert spaces. The latter include graph Laplacians for infinite weighted graphs G, w with the Laplacian in this context presented as a Hermitian operator in an associated Hilbert space of finite energy functions on the vertex set in G.

Other examples include Hilbert spaces of band-limited signals. Further applications enter into the techniques used in discrete simulations of stochastic integrals, see [12]. We encountered the present operator theoretic duality in our study of discrete Laplacians, which in turn have part of its motivation in numerical analysis.

A key tool in applying numerical analysis to solv- ing partial differential equations is discretization, and use of repeated differences; see e. Specifically, one picks a grid size h, and then proceeds in steps: 1 Starting with a partial differential operator, then study an associated discretized operator with the use of repeated differences on the h-lattice in Rd.

Our present approach, based on reproducing kernels and unbounded operators, fits into a larger framework in applied operator theory, for example the use of reproducing kernel Hilbert spaces in the determi- nation of optimal spectral estimation: Here the problem is to estimate some sampled signal represented as the sum of a deterministic time- function and a term representing noise, for example white noise; see e. For the multivariable case, the process under study is in- dexed by some prescribed discrete set X representing sample points; it could be the vertex set in an infinite graph.

The choice of statistical distribution, modeling the noise term, then amounts to a selection of a reproducing kernel representing function differences with vectors vx dipoles in the present context , and linear combinations of these vectors vx in this approach then represents a spectral estimator.

The problem becomes that of selecting samples which minimize error terms in a prediction of a signal. Reproducing kernel-Hilbert spaces For this purpose, one must use a metric, and the norm in Hilbert space has proved an effective tool, hence the Hilbert spaces and the op- erator theory. This procedure connects to our present graph-Laplacians: When discretization is applied to the Laplace operator in d continuous variables, the result is the graph of integer points Zd with constant weights.

But if numerical analysis is applied instead to a continuous Laplace operator on a Riemannian manifold, the discretized Laplace operator will instead involve infinite graph with variable weights, so with vertices in other configurations than Zd. Inside the technical sec- tions we will use standard tools from analysis and probability.

Ref- erences to the fundamentals include [10], [15], [17] and [22]. There is a large literature covering the general theory of reproducing kernel Hilbert spaces and its applications, see e. Such applications include potential theory, stochastic integration, and boundary value problems from PDEs among others. In brief summary, a reproducing kernel Hilbert space consists of two things: a Hilbert space of functions f on a set X, and a reproducing kernel k, i.

Moreover, there is a set of axioms for a function k in two variables that characterizes precisely when it determines a reproducing kernel Hilbert space. And conversely there are necessary and sufficient conditions that apply to Hilbert spaces H and decide when H is a reproducing kernel Hilbert space.

Quantum states in physics are represented by norm-one vectors v in some Hilbert space H, i. When these two additional conditions i — ii are satisfied, we say that H is a relative reproducing kernel Hilbert space. It is known that every weighted graph the infinite case is of main concern here induces a relative reproducing kernel Hilbert space, and an associated graph Laplacian.

A main result in section VII below is that the converse holds: Given a relative reproducing kernel Hilbert space H on a set X, it is then possible in a canonical way to construct a weighted graph G such that X is the set of vertices in G, and such that its energy Hilbert space coincides with H itself.

In our construction, the surprise is that the edges in G as well as the weights on the edges may be built directly from only the Hilbert space axioms defining the initially given relative reproducing kernel Hilbert space.

Since this includes all infinite graphs of electrical resistors and their potential theory boundaries, harmonic functions, and graph Laplacians the result has applications to these fields, and it serves to unify diverse branches in a vast research area. We are con- cerned with harmonic functions h of finite energy, and our reproducing kernel Hilbert spaces are chosen such as to make this precise, as well as serving as a computational device.

This fact further explains why the resulting boundary theory is some- what more subtle than is the better known and better understood the- ory for the case of bounded harmonic functions. In- deed, a fixed choice of weights on edges in G for example conductance numbers yields probabilities for a random walk.

The discreteness of vertex sets in infinite graphs, has a quantum aspect as well [9], [14]. It enters when inner products from a chosen reproducing kernel-Hilbert space is used in encoding transition proba- bilities, i. Hence, vertices in G play the role of quantum states. Let H be a reproducing kernel Hilbert space of functions on some fixed set X; we assume properties i — ii above.

We will see that bipoles always exist, while monopoles do not. Example 3. The corresponding Hilbert space will be denoted H. Lemma 3. The last fact can be verified directly. Proof of Lemma 3. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion.

The rationale for providing an introduction to the theory of Hilbert space, rather than. Spectral Theory of Operators on Hilbert Spaces. This work is a concise introduction to spectral theory of Hilbert space operators. Highly Influenced. View 9 excerpts, cites methods and background. Mathematics, Computer Science. View 4 excerpts, cites background. SIAM J.

View 5 excerpts, cites background. Extremal Dependence Concepts. Computer Science, Mathematics. View 6 excerpts, cites background. Author : J. Paul R. Author : Paul R. Gerald J. Murphy Publisher: Academic Press Reads. Author : Gerald J. Kurt O. Author : Kurt O. Svetlin G. Author : Svetlin G.