A note concerning the index of the shift

Let $\mu$ be a finite, positive Borel measure with support in $\{z: \vert z\vert \leq 1\}$such that $P^2(\mu)$ - the closure of the polynomials in $L^2(\mu)$ - is irreducible and each point in $\mathbb{D} := \{z: \vert z\vert < 1\}$ 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

then $\dim(\mathcal{M}\ominus z\mathcal{M}) = 1$ for each nontrivial closed invariant subspace $\mathcal{M}$ for the shift $M_z$ on $P^2(\mu)$.