Invariant Subspaces and the Exponential Map
نویسندگان
چکیده
Bounded operators with no non-trivial closed invariant subspace have been constructed by P. Enflo [6]. In fact, there exist bounded operators on the space 1 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 nontrivial 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 algebra. 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 Corollary 5 are the only statements of this work which go back to [4].
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