Spectral Theory Of Unbounded Operators Research Articles

Realizations and Factorizations of Positive Definite Kernels

Given a fixed sigma-finite measure space $$\left( X,\mathscr {B},\nu \right) $$ , we shall study an associated family of positive definite kernels K. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $$\nu $$ is infinite, but sigma-finite. We introduce such positive definite kernels $$K\left( \cdot ,\cdot \right) $$ with the two variables from the subclass of the sigma-algebra $$\mathscr {B}$$ whose elements are sets with finite $$\nu $$ measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for K, and we give necessary and sufficient conditions for K to have realizations and factorizations in $$L^{2}\left( \nu \right) $$ . Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals.

Functional analysis and quantum mechanics: an introduction for physicists

We give an introduction to certain topics from functional analysis which are relevant for physics in general and in particular for quantum mechanics. Starting from some examples, we discuss the theory of Hilbert spaces, spectral theory of unbounded operators, distributions and their applications and present some facts from operator algebras. We do not give proofs, but present examples and analogies from physics which should be useful to get a feeling for the topics considered.