  We consider hilbert spaces of holomorphic functions in Cartan domains (in
particular in ball and polydisk) and operator of restriction of holomorphic
function to a submanifold in Shilov boundary. We discuss conditions when this
operator exists.
  Using such 'trace theorems' it is possible to construct discrete increments
to spectra of some unitary representation and to catch singular unitary
representations in the spectra. We also discuss spectral problems related to
Berezin type kernels on riemann symmetric spaces.
