Completely indecomposable operators and a uniqueness theorem of Cartwright-Levinson type
A completely indecomposable operator is a bounded linear operator \(T\) on some complex Hilbert space, such that the spectrum of \(T\) is not a singleton, is contained in the spectrum of the restriction of \(T\) to any nonzero invariant subspace and the analogous property holds for \(T^*\). For example, neither the unilateral shift, nor the bilateral one is completely indecomposable. The aim of the paper under review is to provide two classes of completely indecomposable operators. The first class consists of self-adjoint operators with spectrum \([-2,2]\) perturbed by certain compact quasinilpotent operators. At present it is not known whether or not these operators actually possess nontrivial invariant subspaces. The operators from the second class are perturbations of the bilateral shift by certain compact quasinilpotent operators and their spectrum is the unit circle. Concerning these operators, the question arises whether or not they possess nontrivial hyperinvariant subspaces. The proofs of complete indecomposability for the above mentioned operators involve local spectral theory, hence analytic extensions of the resolvent. The main technical ingredient here is a new uniqueness theorem of Cartwright-Levinson type, concerning analytic extension of holomorphic functions in the complement of the unit circle. We should finally note that, besides of its extremely important theorems, the paper also contains several open problems and additional discussions.
scientific article; zbMATH DE number 1398377