Unitary quasi-affine transforms of contractions

The unitary quasi-affine transforms of contractions are the cornerstone of the exact theory of irreversibility introduced by Misra, Prigogine and Courbage. This work shows that the class of contractions with unitary quasi-affine transforms is just C ⋅ 1 and that every unitary quasi-affine transform of a contraction is unitarily equivalent to the residual part of its Sz.-Nagy–Foiaş dilation. Some connections with the Grossmann theory of nested Hilbert spaces and generalized eigenvalues are established. These results are applied to the study of the baker map.