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arXiv · 2111.09249

Higher Rank Numerical Ranges and Unitary Dilations

Pankaj Dey·Mithun Mukherjee
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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.

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Pankaj Dey, Mithun Mukherjee. 2021-11-17. Higher Rank Numerical Ranges and Unitary Dilations. https://doi.org/10.1007/s43036-022-00217-x

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