The polynomial numerical hulls of Jordan blocks and related matrices

The polynomial numerical hull of degree k for a square matrix A is a set designed to give useful information about the norms of polynomial functions of the matrix; it is defined as {z C: p(A) p(z) forall p ofdegree k orless}.Whilethese sets have been computed numerically for a number of matrices, the computations have not been verified analytically in most cases. In this paper we show analytically that the 2-norm polynomial numerical hulls of degrees 1 through n−1 for an n by n Jordan block are disks about the eigenvalue with radii approaching 1 as n→∞, and we prove a theorem characterizing these radii rk,n. In the special case where k=n−1, this theorem leads to a known result in complex approximation theory: For n even, rn−1,n is the positive root of 2rn+r−1=0, and for n odd, it satisfies a similar formula. For large n, this means that rn−1,n≈1−log(2n)/n+log(log(2n))/n. These results are used to obtain bounds on the polynomial numerical hulls of certain degrees for banded triangular Toeplitz matrices and for block diagonal matrices with triangular Toeplitz blocks.