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

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.