A model for operators with cyclic adjoint
type: Item
instance of: scholarly article
title: A model for operators with cyclic adjoint
zbMATH Open document ID: 0623.47006
DOI: 10.1007/BF01199076
author: Aharon Atzmon
published in: Integral Equations and Operator Theory
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47A45
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 46E20
zbMATH DE Number: 4011231
zbMATH Keywords: weak-star-cyclic vector
zbMATH Keywords: differentiation operator on a Banach space of entire functions of exponential type
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: similarity
zbMATH Keywords: unitary equivalence
zbMATH Keywords: translation group
zbMATH Keywords: invariant subspaces
zbMATH Keywords: translation invariant subspaces
reviewed by: Joseph A. Ball
MaRDI profile type: Publication
cites work: Operators which are annihilated by analytic functions and invariant subspaces
cites work: An operator without invariant subspaces on a nuclear Fréchet space
cites work: Un opérateur sans sous-espace invariant: Simplification de l'exemple de P. Enflo
cites work: Q5833361
cites work: Q4108810
cites work: Cyclic Vectors of a Cyclic Operator Span the Space
cites work: A Solution to the Invariant Subspace Problem
cites work: Q4726901
cites work: The existence of invariant subspaces
cites work: Q3874718
description: scientific article; zbMATH DE number 4011231