Abstract
We describe the relationship between the expansion of a self-adjoint operator in generalized eigenvectors and the direct integral of Hilbert spaces. We perform the explicit diagonalization of a self-adjoint absolutely continuous singular integral operator Y using an Hermitian nonnegative kernel consisting of boundary values of the determining function of the operator T = X + iY with respect to the resolvent of the imaginary part of Y.
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Vorob'ev, I.V. Expansion of a Self-Adjoint Absolutely Continuous Singular Integral Operator in Generalized Eigenvectors and Its Diagonalization. Ukrainian Mathematical Journal 55, 171–179 (2003). https://doi.org/10.1023/A:1025089122778
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DOI: https://doi.org/10.1023/A:1025089122778