  Krein-de Branges spectral theory establishes a correspondence between the
class of differential operators called canonical Hamiltonian systems and
measures on the real line with finite Poisson integral. We further develop this
area by giving a description of canonical Hamiltonian systems whose spectral
measures have logarithmic integral converging over the real line. This result
can be viewed as a spectral version of the classical Szego theorem in the
theory of polynomials orthogonal on the unit circle. It extends Krein-Wiener
completeness theorem, a key fact in the prediction of stationary Gaussian
processes.
