Near invariance and symmetric operators

2012/07/12 by R. T. W. Martin, Martin, R. T. W.
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1207.2931

arxiv created 2012/07/12 · arxiv updated 2012/07/13

Abstract

Let S be a subspace of L² (\bmR). We show that the operator M of multiplication by the independent variable has a simple symmetric regular restriction to S with deficiency indices (1,1) if and only if S = u h K²_θ is a nearly invariant subspace, with θ a meromorphic inner function vanishing at i. Here u is unimodular, h is an isometric multiplier of K²_θ into H² and H² is the Hardy space of the upper half plane. Our proof uses the dilation theory of completely positive maps.

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