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 is an element of C: parallel top(A)parallel to greater than or equal to \p(z)\ for all p of degree k or less}. While these 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 I through n-1 for an n by n Jordan block are disks about the eigenvalue with radii approaching I as n --> infinity, and we prove a theorem characterizing these radii r(k,n). In the special case where k = n-1, this theorem leads to a known result in complex approximation theory: For n even, r(n-1,n) is the positive root of 2r(n) + r-1 = 0, and for n odd, it satisfies a similar formula. For large n, this means that r(n-1,n) approximate to 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. (C) 2003 Elsevier Inc. All rights reserved.