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arXiv · 1801.04915

Phillips symmetric operators and their extensions

S. Kuzhel·L. Nizhnik
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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}.

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S. Kuzhel, L. Nizhnik. 2018-01-12. Phillips symmetric operators and their extensions. https://doi.org/10.1215/17358787-2018-0009

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