Title: Quasianalytic contractions and function algebras

Authors: Laszlo Kerchy

Issue: Volume 60 (2011), Issue 1, 21-40


Completing former results in [L. K\'erchy, \textit{On the hyperinvariant subspace problem for asymptotically nonvanishing contractions}, Operator Theory Adv. Appl., \textbf{127} (2001), 399--422], the effect of the Sz.-Nagy--Foias functional calculus on the unitary asymptote of a contraction is described. The hyperinvariant subspace problem for a class of cyclic, quasianalytic $C_{10}$-contractions is reduced to the particular case, when the quasianalytic spectral set coincides with the unit circle $\mathbb{T}$. In this setting the commutant $\{T\}$ of $T$ is identified with a quasianalytic subalgebra $\mathcal{F}(T)$ of $L^{\infty}(\mathbb{T})$ containing $H^{\infty}$. Conditions are given for the cases when $\mathcal{F}(T)$ is a Douglas algebra, a pre-Douglas algebra, or a generalized Douglas algebra.