Phillips symmetric operators and their extensions

TL;DR

This paper studies Phillips symmetric operators, which are invariant under a set of unitaries and have constant characteristic functions, highlighting their unique extension properties in Hilbert spaces.

Contribution

It characterizes Phillips symmetric operators with constant characteristic functions and explores their extension properties and invariance features.

Findings

Existence of symmetric operators with no invariant self-adjoint extensions

Constant characteristic function implies Phillips symmetric operator

Invariance under a set of unitaries influences extension theory

Abstract

Let $S$ be a symmetric operator with equal defect numbers and let $\mathfrak{U}$ be a set of unitary operators in a Hilbert space $\mathfrak{H}$. The operator $S$ is called $\mathfrak{U}$-invariant if $US=SU$ for all $U\in\mathfrak{U}$. Phillips \cite{PH} constructed an example of $\mathfrak{U}$-invariant symmetric operator $S$ which has no $\mathfrak{U}$-invariant self-adjoint extensions. It was discovered that such symmetric operator has a constant characteristic function \cite{KO}. For this reason, each symmetric operator $S$ with constant characteristic function is called a \emph{Phillips symmetric operator}.