Higher Rank Numerical Ranges and Unitary Dilations

TL;DR

This paper extends finite-dimensional results on the relationship between the k-rank numerical range of contractions and their unitary dilations to infinite-dimensional Hilbert spaces, with specific conditions and counterexamples.

Contribution

It generalizes a finite-dimensional theorem to infinite dimensions, showing the closure of the k-rank numerical range equals the intersection over all unitary dilations, and explores related C-numerical range properties.

Findings

Closure of k-rank numerical range equals intersection over unitary dilations.

Extension of finite-dimensional results to infinite-dimensional spaces.

Counterexample for C-numerical range case.

Abstract

Here we show that for $k\in \mathbb N,$ the closure of the $k$-rank numerical range of a contraction $A$ acting on an infinite-dimensional Hilbert space $\mathcal{H}$ is the intersection of the closure of the $k$-rank numerical ranges of all unitary dilations of $A$ to $\mathcal{H}\oplus\mathcal{H}.$ The same is true for $k=\infty$ provided the $\infty$-rank numerical range of $A$ is non-empty. These generalize a finite dimensional result of Gau, Li and Wu. We also show that when both defect numbers of a contraction are equal and finite ($=N$), one may restrict the intersection to a smaller family consisting of all unitary $N$-dilations. A result of {Bercovici and Timotin} on unitary $N$-dilations is used to prove it. Finally, we have investigated the same problem for the $C$-numerical range and obtained the answer in negative.