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Let μ be a finite, positive Borel measure with support in {z: |z| 0 and there is a nontrivial subarc γ of ∂D such that log()dm > -∞, then dim(M ⊖ zM) = 1 for each nontrivial closed invariant subspace M for the shift M z on P 2 (μ).

A note concerning the index of the shift