Spectral Theorem for Self-Adjoint Partial Integral Operators in Kaplansky-Hilbert Modules

TL;DR

This paper proves a spectral theorem for self-adjoint partial integral operators in Kaplansky-Hilbert modules, extending classical results to a more general functional analytic setting.

Contribution

It introduces a spectral theorem for self-adjoint cyclically compact operators in Kaplansky-Hilbert modules, generalizing Mercer’s theorem for positive definite kernels.

Findings

Spectral decomposition via eigenfunctions established.

Integral representation with orthogonal projectors demonstrated.

Functional calculus for these operators developed.

Abstract

In this paper, a spectral theorem is proved for self-adjoint cyclically compact partial integral operators in the space of functions with mixed norm, which is a Kaplansky--Hilbert module. The decomposition through eigenfunctions, integral representation using orthogonal projectors, and functional calculus are established. The results generalize Mercer theorem for positive definite kernels. The proofs rely on the gluing of projector-valued measures, presented in separate lemmas. An example illustrates all assertions of the theorem for a specific kernel and function.