Zero-dilation Indices of KMS Matrices

Abstract

The zero-dilation index d(A) of an n-by-n complex matrix A is the maximum size of the zero matrix which can be dilated to A. In this paper, we determine the value of this index for the KMS matrix * * . Hence the zero-dilation index of A can

Blaschke Products, Level Sets, and Crouzeix's Conjecture

“…The matrix A t is sometimes called a KMS matrix and the numerical ranges of these matrices have been studied by Gau and Wu in [27,28]. In the 3 × 3 case, Crouzeix's results from [14] can be applied to obtain the boundary of W (A t ) and hence, of W (M t ).…”

“…The statement of ( 2) suggests that we need to study the numerical range of the associated compression of the shift S Θ . For unicritical Θ, these numerical ranges were studied by Gaaya in [21,22], Gau and Wu in [27,28], and in work of Partington and the second author [30]. When deg Θ = 3, results about W (S Θ ) are encoded in Crouzeix's work [14]; indeed, his arguments imply that the full Crouzeix conjecture holds for S Θ when deg Θ = 3 and Θ is unicritical.…”