Set theory and cyclic vectors

Abstract

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 κis a regular cardinal such that κ\geq ω_1 and 2^{<κ} = κ. Then it is relatively consistent with ZFC that 2^ω= κand for any subset S \subset H of cardinality less than κthe 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.