Trace formula for contractions and its representation in d

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 a 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.