On the Theory of Generalized Toeplitz Kernels

A new proof of the integral representation of the generalized Toeplitz kernels is given. This proof is based on the spectral theory of the corresponding differential operator that acts in the Hilbert space constructed from a kernel of this sort. A theorem on conditions that should be imposed on the kernel to guarantee the self-adjointness of the operator considered (i.e., the uniqueness of the measure in the representation) is proved.