Eigenvector bases of completely nonunitary contractions and the characteristic function

We study Bari and Riesz bases $({}^1)$ of eigenspaces of contraction operators which are close to unitary. Subject to certain assumptions about the operator, we partition its spectrum into so-called Carleson series, in terms of which we establish new criteria for the basicity of the operator. Most completely studied are contractions with finite-dimensional deficiency operators $I-T^*T$ and $I-TT^*$. As examples we consider classical bases of exponential functions in various function spaces.