Spectrum of the Sum of Partial Integral Operators Generated by Incomplete Orthonormal Systems
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In this paper, we study a class of partial integral operators acting in the Hilbert space L 2 (Ω 1 ×Ω 2 ),
generated by incomplete orthonormal systems in the corresponding L 2 spaces. Using a direct integral
decomposition, we obtain an explicit description of the spectra of these operators in terms of the
essential ranges of the generating coefficient functions. It is shown that the incompleteness of the
underlying orthonormal systems leads to the appearance of an additional spectral component at zero.
The main result concerns the spectral analysis of the sum of two such partial integral operators. We
provide a precise characterization of the spectrum of the sum operator by exploiting its fiber-wise
structure. The obtained results contribute to the spectral theory of non-compact partial integral
operators and can be applied to related problems in operator theory and mathematical physics.
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