A model for operators with cyclic adjoint
The author shows that every bounded linear operator on a complex Banach space whose adjoint has a weak-star-cyclic vector, is similar to the differentiation operator on a Banach space of entire functions of exponential type. In the case that the Banach space is a Hilbert space, the space of entire functions is a reproducing kernel Hilbert space and the similarity is a unitary equivalence. The class of Banach spaces of entire functions invariant under the differentiation operator for which the differentiation operator is bounded is characterized. In particular such a Banach space E is invariant under the translation operator \((S_ zf)(w)=f(w+z)\) and the translation group \(z\to S_ z\) is strongly continuous. Recent negative results on invariant subspaces for general Banach space operators due to P. Enflo and C. J. Read lead to the conclusion that there exists such a Banach space of entire functions which has no proper closed translation invariant subspaces. The open invariant subspace problem for Hilbert space operators is thus reduced to a harmonic analysis question concerning the existence of translation invariant closed subspaces in an admissible Hilbert space of entire functions.
differentiation operator on a Banach space of entire functions of exponential type
scientific article; zbMATH DE number 4011231