Self-adjoint cyclically compact operators and their applications

TL;DR

This paper studies self-adjoint cyclically compact operators on Hilbert--Kaplansky modules, establishes their spectral theorem, and applies these results to partial integral equations and compact operators in von Neumann algebras.

Contribution

It introduces a spectral theorem for self-adjoint cyclically compact operators and applies it to solve partial integral equations and characterize compact operators in von Neumann algebras.

Findings

Spectral theorem for self-adjoint cyclically compact operators.

Condition for solvability of partial integral equations.

General form of compact operators relative to type I von Neumann algebras.

Abstract

This paper is devoted to self-adjoint cyclically compact operators on Hilbert--Kaplansky module over a ring of bounded measurable functions. The spectral theorem for such a class of operators are given. We apply this result to partial integral equations on the space with mixed norm of measurable functions and to compact operators relative to von Neumann algebras. We will give a condition of solvability of partial integral equations with self-adjoint kernel. Moreover, a general form of compact operators relative to a type I von Neumann algebra is given.