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.
Citation Example:Vorob'ev I. V. Expansion of a Self-Adjoint Absolutely Continuous Singular Integral Operator in Generalized Eigenvectors and Its Diagonalization // Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 138-145.