Higher Rank Numerical Ranges of Normal Operators and unitary dilations

TL;DR

This paper characterizes the higher rank numerical range of normal operators on infinite-dimensional spaces using spectral measures, generalizes existing results, and explores relationships with unitary dilations and contractions.

Contribution

It extends the description of higher rank numerical ranges to infinite-dimensional normal operators and establishes conditions relating these ranges to unitary dilations.

Findings

Spectral measure characterizes higher rank numerical range of normal operators.

Existence of normal contractions with proper subset relations in their higher rank numerical ranges.

Necessary and sufficient conditions for equality of higher rank numerical ranges and their dilations.

Abstract

We describe here the higher rank numerical range, as defined by Choi, Kribs and Zyczkowski, of a normal operator on an infinite dimensional Hilbert space in terms of its spectral measure. This generalizes a result of Avendano for self-adjoint operators. An analogous description of the numerical range of a normal operator by Durszt is derived for the higher rank numerical range as an immediate consequence. It has several interesting applications. We show using Durszt's example that there exists a normal contraction $T$ for which the intersection of the higher rank numerical ranges of all unitary dilations of $T$ contains the higher rank numerical range of $T$ as a proper subset. Finally, we strengthen and generalize a result of Wu by providing a necessary and sufficient condition for the higher rank numerical range of a normal contraction being equal to the intersection of the higher rank numerical ranges of all possible unitary dilations of it.