Spectral measures and duality of spectral subspaces of contractions with slowly growing resolvent

We consider the class $\Pi$ of contracting operators $T$ with spectrum on the unit circle $\Gamma$, acting on a separable Hilbert space and subject to the following restriction on the growth of the resolvent $R_T(\lambda)$:
$$ \sup_{0\leqslant\rho<1}\int^{2\pi}_0\ln^+\{(1-\rho)\|R_T(\rho e^{i\varphi})\|\}d\varphi<+\infty. $$
We study the spectral subspaces $\Omega_T(B)$ for $T\in\Pi$, corresponding to arbitrary Borel subsets of the circle $\Gamma$; in parallel we study a Borel measure $\omega_T(B)$ on $\Gamma$, adequate for $\Omega_T(B)$ in the following sense:
$$ \Omega_T(B)=\{0\}\Longleftrightarrow\omega_T(B)=0. $$