A note concerning the index of the shift

Abstract:

Let $\mu$ be a finite, positive Borel measure with support in $\{z: |z| \leq 1\}$ such that $P^2(\mu )$ – the closure of the polynomials in $L^2(\mu )$ – is irreducible and each point in $\mathbb {D} := \{z: |z| < 1\}$ is a bounded point evaluation for $P^2(\mu )$. We show that if $\mu (\partial {\mathbb {D}}) > 0$ and there is a nontrivial subarc $\gamma$ of $\partial {\mathbb {D}}$ such that \[ \int _{\gamma }log(\mbox {\small {$\frac {d\mu }{dm}$}})dm > -\infty ,\] then $\dim (\mathcal {M}\ominus z\mathcal {M}) = 1$ for each nontrivial closed invariant subspace $\mathcal {M}$ for the shift $M_z$ on $P^2(\mu )$.