On singular parts of contractive analytic operator-functions

We consider the class $B_G$ of holomorphic functions in $G\in\mathbb C$ whose values are contractions on a separable Hilbert space. For $T(\cdot)\in B_G$ we prove that if $T(z_0)$ (for some $z_0\in G$) is a weak contraction, its singular part $T^{(s)}(z_0)$ is complete, and the difference $T(z)-T(z_0)$ is not too big (say, finite dimensional) then $T^{(s)}(z_0)$ is complete almost everywhere in $G$. If, in addition, $T(z_0)$ is a completely nonunitary contraction satisfying some smoothness conditions then the spectrum $\sigma_z$ of $T^{(s)}(z_0)$ $(z\in G)$ is a thin set (in nontrivial case):
$$ \int_{\mathbb T}\log\{\inf_{\zeta\in\sigma_z}|t-\zeta|\}\,|dt|>-\infty. $$
The proofs of the results stated are based on a formula obtained in the paper which relates the characteristic functions of the contractions $T(z)$ for different $z$ in $G$.