Higher Rank Numerical Ranges and Unitary Dilations

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.