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A note concerning the index of the shift
Authors:John R. Akeroyd
Affiliation:Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
Abstract:Let $mu$ be a finite, positive Borel measure with support in ${z: vert zvert leq 1}$such that $P^2(mu)$ - the closure of the polynomials in $L^2(mu)$ - is irreducible and each point in $mathbb{D} := {z: vert zvert < 1}$ 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

begin{displaymath}int_{gamma}log(mbox{small {$frac{dmu}{dm}$ }})dm > -infty,end{displaymath}

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

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