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

Yu. P. Ginzburg

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 203-206

Résumé

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: $$ \Omega_T(B)=\{0\}\Longleftrightarrow\omega_T(B)=0. $$