On the Mohebi-Radjabalipour conjecture. (Q1862720)

Subdecomposable operators with thick spectra have nontrivial invariant subspaces according to a result of \textit{J.~Eschmeier} and \textit{B.~Prunaru} [J. Funct. Anal. 94, 196--222 (1990; Zbl 0744.47003)]. In this connection, \textit{H.~Mohebi} and \textit{M.~Radjabalipour} [Integral Equations Oper. Theory 18, 222--241 (1994; Zbl 0807.47020)] asked whether this result can be generalized for operators \(T\) that have extensions which, although not necessarily decomposable, still have a rich spectral behavior on an open set where the spectrum of \(T\) is dominating. In the paper under review, the author answers this question in a special case that covers, in particular, the invariant subspace theorems of \textit{S. W.~Brown} [Integral Equations Oper. Theory 1, 310--333 (1978; Zbl 0416.47009)] and \textit{C.~Apostol} [J. Oper. Theory 6, 3--12 (1981; Zbl 0479.47003)].