Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Universal vectors for operators on spaces of holomorphic functions

Authors: Robert M. Gethner and Joel H. Shapiro
Journal: Proc. Amer. Math. Soc. 100 (1987), 281-288
MSC: Primary 47B38; Secondary 30D20, 30H05
MathSciNet review: 884467
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Abstract: A vector $ x$ in a linear topological space $ X$ is called universal for a linear operator $ T$ on $ X$ if the orbit $ \{ {T^n}x:n \geq 0\} $ is dense in $ X$. Our main result gives conditions on $ T$ and $ X$ which guarantee that $ T$ will have universal vectors. It applies to the operators of differentiation and translation on the space of entire functions, where it makes contact with Pólya's theory of final sets; and also to backward shifts and related operators on various Hilbert and Banach spaces.

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Keywords: Cyclic vector, entire function, backward shift, final set, successive derivatives
Article copyright: © Copyright 1987 American Mathematical Society