  A relation between values of a unitarily invariant norm of Hermitian operator
before and after action of completely positive map is studied. If the norm is
jointly defined on both the input and output Hilbert spaces, one defines a
shrinking factor under the restriction of given map to Hermitian operators. As
it is shown, for any unitarily invariant norm this shrinking factor is not
larger than the maximum of two values for the spectral norm and the trace norm.
