  The aim of this article is twofold: give a short proof of the existence of
real spectral shift function and the associated trace formula for a pair of
contractions, the difference of which is trace-class and one of the two a
strict contraction, so that the set of assumptions is minimal in comparison to
those in all the existing proofs. The second one is to find a trace formula for
differences of functions of contraction and its adjoint, in which case, the
integral in the formula is over the unit disc and has an expression
surprisingly similar to the Helton-Howe formula.
