$P^{2}(\mu )$ and bounded point evaluations
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- by Tavan T. Trent and James L. Wang PDF
- Proc. Amer. Math. Soc. 91 (1984), 421-425 Request permission
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.References
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Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 421-425
- MSC: Primary 47B38; Secondary 46J10
- DOI: https://doi.org/10.1090/S0002-9939-1984-0744642-3
- MathSciNet review: 744642