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The aim of this article is twofold: give a short proof of the existence ofreal spectral shift function and the associated trace formula for a pair ofcontractions, the difference of which is trace-class and one of the two astrict contraction, so that the set of assumptions is minimal in comparison tothose in all the existing proofs. The second one is to find a trace formula fordifferences of functions of contraction and its adjoint, in which case, theintegral in the formula is over the unit disc and has an expressionsurprisingly similar to the Helton-Howe formula.

Trace formula for contractions and it's representation in $\mathbb{D}$