Contractions with spectral boundary
The purpose of this paper is to prove the following result: If T is a contraction on a complex, separable, infinite-dimensional Hilbert space such that T \(n\to 0\) and \(T^{*n}\to 0\) strongly and \(\sigma\) (T) contains the unit circle, then T has a nontrivial invariant subspace. This is proved by realizing T in its functional model and showing that the predual of the ultraweakly closed algebra generated by T and I consists entirely of rank-one linear functionals. [For the dual algebra theory, consult \textit{H. Bercovici}, \textit{C. Foiaş} and \textit{C. Pearcy}'s, Dual algebras with applications to invariant subspaces and dilation theory, Reg. Conf. Ser. Math. 56, 108 p. (1985; Zbl 0569.47007)]. More recently, there are further progresses along this line of developments. It is now known that (1) every contraction T on a Hilbert space with \(\sigma\) (T) containing the unit circle has a nontrivial invariant subspace [cf. \textit{S. W. Brown}, \textit{B. Chevreau} and \textit{C. Pearcy}, J. Func. Anal. 76, 30-55 (1988; Zbl 0641.47013)], and, more generally, (2) every contraction T with \(\sigma\) (T) containing the unit circle is either reflexive or has a nontrivial hyperinvariant subspace [cf. \textit{B. Chevreau}, \textit{G. Exner} and \textit{C. Peary}, Structure theory and reflexivity of contraction operators, Bull. Am. Math. Soc., New Ser. 19, 299-301 (1988)].
scientific article; zbMATH DE number 4063842
scientific article