Orbital and strongly orbital spaces

详细信息    查看全文

• 作者：Stanislav Shkarin
• 关键词：Cyclic operators ; Hypercyclic operators ; Invariant subspaces ; Topological vector spaces
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：15 March, 2014
• 年：2014
• 卷：411
• 期：2
• 页码：794-815
• 全文大小：526 K

文摘

We say that a (countably dimensional) topological vector space is orbital if there is and a vector such that is the linear span of the orbit . We say that is strongly orbital if, additionally, can be chosen to be a hypercyclic vector for . Of course, can be orbital only if the algebraic dimension of is finite or infinite countable. We characterize orbital and strongly orbital metrizable locally convex spaces. We also show that every countably dimensional metrizable locally convex space does not have the invariant subset property. That is, there is such that every non-zero is a hypercyclic vector for . Finally, assuming the Continuum Hypothesis, we construct a complete strongly orbital locally convex space.

As a byproduct of our constructions, we determine the number of isomorphism classes in the set of dense countably dimensional subspaces of any given separable infinite dimensional Fr茅chet space . For instance, in , there are exactly 3 pairwise non-isomorphic (as topological vector spaces) dense countably dimensional subspaces.