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.

Contents Full-Text PDF