Journal of Operator Theory

Volume 52, Issue 1, Summer 2004 pp. 133-138.

Set theory and cycle vectors

Authors: Nik Weaver
Author institution: Department of Mathematics, Washington University, St. Louis, MO 63130, USA

Summary: Let $H$ be a separable, infinite dimensional Hilbert space and let $S$ be a countable subset of $H$. Then most positive operators on $H$ have the property that every nonzero vector in the span of $S$ is cyclic, in the sense that the set of operators in the positive part of the unit ball of $B(H)$ with this property is comeager for the strong operator topology. Suppose $\kappa$ is a regular cardinal such that $\kappa \geq \omega_1$ and $2^{<\kappa} = \kappa$. Then it is relatively consistent with ZFC that $2^\omega = \kappa$ and for any subset $S \subset H$ of cardinality less than $\kappa$ the set of positive operators in the unit ball of $B(H)$ for which every nonzero vector in the span of $S$ is cyclic is comeager for the strong operator topology.

Keywords: cyclic vector, invariant subspace, Martin's axiom

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