Full analytic subspaces for contractions with rich spectrum (Q1105819)

Let S be a completely nonunitary contraction on a complex, separable Hilbert space. Assume that the left essential spectrum \(\sigma_{1e}(S)\) of S is dominating in the unit disc, meaning that almost every point of the unit circle is a nontangential limit of a sequence from \(\sigma_{1e}(S)\). (S is called a (BCP)-operator by some previous authors.) The main result of this paper gives that S has a ``full analytic'' invariant subspace. The precise definition of ``full analytic'' is too technical to be given here. It \(g(x,y)\in D(T^ 3)\), and \(x=x(t)\), \(y=y(t)\) is a parametrization of the boundary of \(\Omega\). The present article is a generalization (to higher orders of the operators) of an article of \textit{V. A. Sadovnichij} and \textit{V. A. Lyubishkin} [Funkts. Anal. Prilozh. 20, No.3, 55-65, 96 (1986)].