Near invariance and symmetric operators

Let $S$ be a subspace of $L^2 (\bm{R})$. 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 $S = u h K^{2}_θ$ is a nearly invariant subspace, with $θ$ a meromorphic inner function vanishing at $i$. Here $u$ is unimodular, $h$ is an isometric multiplier of $K^{2}_θ$ into $H^2$ and $H^2$ is the Hardy space of the upper half plane. Our proof uses the dilation theory of completely positive maps.

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