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<p>The paper is devoted to the study of the spectrum of a self-adjoint bounded linear operator on a Kaplansky-Hilbert module over a ring of measurable functions. Using the technique of measurable bundles, we show that an arbitrary Kaplansky-Hilbert module over this ring can be represented as a measurable bundle of Hilbert spaces. Furthermore, we prove that the cyclic modular spectrum of a self-adjoint operator on this Kaplansky-Hilbert module can be expressed as a measurable bundle of the spectra of bounded linear operators acting on Hilbert spaces. We also present an application of this representation to the solvability of partial integral equations.</p>

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ON THE SPECTRUM OF A SELF-ADJOINT OPERATOR ON KAPLANSKY-HILBERT MODULES

  • 期刊论文

  • 发表日期：

    2025年4月29日

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  • Karimbergen Kudaybergenov,
  • Allabay Arziev,
  • Parakhatdiin Orinbaev

摘要

The paper is devoted to the study of the spectrum of a self-adjoint bounded linear operator on a Kaplansky-Hilbert module over a ring of measurable functions. Using the technique of measurable bundles, we show that an arbitrary Kaplansky-Hilbert module over this ring can be represented as a measurable bundle of Hilbert spaces. Furthermore, we prove that the cyclic modular spectrum of a self-adjoint operator on this Kaplansky-Hilbert module can be expressed as a measurable bundle of the spectra of bounded linear operators acting on Hilbert spaces. We also present an application of this representation to the solvability of partial integral equations.