Invariant Subspaces and the Exponential Map

Invariant Subspaces and the Exponential Map Positivity 8: 101–107, 2004. 101 © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 1 2 3 A. ATZMON , G. GODEFROY and N. J. KALTON School of Mathematical Analysis, Tel-Aviv University, Tel-Aviv 69978, Israel. (E-mail:; Equipe d’Analyse, Université Paris 6, Case 186, 4, Place Jussieu, 75252 Paris Cedex 05, France. (E-mail:; University of Missouri-Columbia, Department of Mathematics, Columbia, MO 65211, USA. (E-mail: (Received 20 September 2001; accepted 10 February 2003) Bounded operators with no non-trivial closed invariant subspace have been con- structed by P. Enflo [6]. In fact, there exist bounded operators on the space with no non-trivial closed invariant subset [12]. It is still unknown, however, if such operators exist on reflexive Banach spaces, or on the separable Hilbert space. The main result of this note (Theorem 1) asserts that the existence of an invariant non- trivial closed subset for the image of an algebra under the exponential map implies the existence of an invariant non-trivial closed subspace for the operators in the al- gebra. The proof relies on a simple differentiation argument. Several consequences of the main result are gathered. This work relies in part on the Note [4]. However, Corollary 2 and Positivity Springer Journals

Invariant Subspaces and the Exponential Map

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Kluwer Academic Publishers
Copyright © 2004 by Kluwer Academic Publishers
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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