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

2025/05/19 by Karimbergen Kudaybergenov, Kudaybergenov, K., Allabay Arziev +3
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.14837

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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