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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cyclic vectors in the Dirichlet space


Authors: Leon Brown and Allen L. Shields
Journal: Trans. Amer. Math. Soc. 285 (1984), 269-303
MSC: Primary 30H05; Secondary 46E99, 47B38
MathSciNet review: 748841
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Abstract: We study the Hilbert space of analytic functions with finite Dirichlet integral in the open unit disc. We try to identify the functions whose polynomial multiples are dense in this space. Theorems 1 and 2 confirm a special case of the following conjecture: if $ \vert f(z)\vert \geqslant \vert g\,(z)\vert$ at all points and if $ g$ is cyclic, then $ f$ is cyclic. Theorems 3-5 give a sufficient condition ($ f$ is an outer function with some smoothness and the boundary zero set is at most countable) and a necessary condition (the radial limit can vanish only for a set of logarithmic capacity zero) for a function $ f$ to be cyclic.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0748841-0
PII: S 0002-9947(1984)0748841-0
Keywords: Banach spaces of analytic functions, Dirichlet integral, weak invertibility, linear operator, invariant subspace, logarithmic capacity
Article copyright: © Copyright 1984 American Mathematical Society