集合论与循环向量
Set theory and cyclic vectors
摘要：
设H是一个可分的无限维希尔伯特空间，设S是H的一个可数子集。则H上的大多数正算子都具有S的跨距中的每个非零向量都是循环的性质，在这个意义上，B（H）的单位球的正部分中具有这一性质的算子集是强算子拓扑的集合.设κ是正则基数使得κ \geq ω_1和2^{<κ} = κ。则2^ω= κ与ZFC是相对一致的，且对基数小于κ的子集S \subset H，B（H）的单位球中的正算子集对强算子拓扑是一致的.
英文摘要：
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.
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