Characterizations of the polynomial numerical hull of degree \(k\)
For bounded linear operators on a Hilbert space, six characterizations of the polynomial numerical hull of degree~\(k\) are established. It is shown how these characterizations provide a natural distinction between interior and boundary points. One of the characterizations is used to prove that the polynomial numerical range of any fixed degree \(k\) for a Toeplitz matrix whose symbol is piecewise continuous and approaches all or most of the infinite-dimensional Toeplitz operators, as the matrix size goes to infinity. More precisely, it is shown that the polynomial numerical hull of degree \(k\) for the Toeplitz operator contains the uniform and partial limits as \(N\to \infty\) of the polynomial numerical hulls of degree \(k\) for the \(N\times N\) Toeplitz matrix, which in turn contain the closure of the interior of that of the Toeplitz operator.
scientific article; zbMATH DE number 5077725