Near invariance and symmetric operators

TL;DR

This paper characterizes nearly invariant subspaces of the Hardy space where the multiplication operator has a symmetric restriction with deficiency indices (1,1), using dilation theory of completely positive maps.

Contribution

It provides a complete characterization of subspaces with symmetric restrictions of the multiplication operator, linking them to nearly invariant subspaces and inner functions.

Findings

Characterization of subspaces where the multiplication operator has symmetric restriction with deficiency indices (1,1)

Connection between nearly invariant subspaces and symmetric operators in Hardy spaces

Use of dilation theory of completely positive maps in the proof

Abstract

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 $S = u h K^{2}_\theta$ is a nearly invariant subspace, with $\theta$ a meromorphic inner function vanishing at $i$. Here $u$ is unimodular, $h$ is an isometric multiplier of $K^{2}_\theta$ 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.