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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

$ P\sp{2}(\mu )$ and bounded point evaluations


Authors: Tavan T. Trent and James L. Wang
Journal: Proc. Amer. Math. Soc. 91 (1984), 421-425
MSC: Primary 47B38; Secondary 46J10
MathSciNet review: 744642
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Abstract: It is shown that if $ g \in C_c^1({\mathbf{C}})$ with $ \bar \partial g$ nonvanishing on the support of $ \mu $ and if $ {P^2}(\mu )$ has no bounded point evaluations, then $ {\text{sp}}{\{ {P^2}(\mu ) + g{P^2}(\mu )\} ^ - } = {L^2}(\mu )$. Similar theorems stating that in the absence of bounded point evaluations $ {P^2}(\mu )$ is "almost" $ {L^2}(\mu )$ are derived. As a consequence, to show that $ {P^2}(\mu ) = {L^2}(\mu )$ in the absence of bounded point evaluations, one need only show that, for example, $ \sqrt {z - \lambda } \in {P^2}(\mu )$ for complex $ \lambda $'s.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0744642-3
PII: S 0002-9939(1984)0744642-3
Keywords: $ {P^2}(\mu )$, bounded point evaluation
Article copyright: © Copyright 1984 American Mathematical Society