Crouzeix's conjecture holds for tridiagonal 3x3 matrices with elliptic numerical range centered at an eigenvalue

D4 Published development or research report or study

Internal Authors/Editors

Publication Details

List of Authors: Christer Glader, Mikael Kurula, Mikael Lindström
Publication year: 2017


M. Crouzeix formulated the following conjecture in (Integral Equations
Operator Theory 48, 2004, 461--477): For every square matrix A and every
polynomial p,

∥p(A)∥≤2maxz∈W(A)|p(z)|, where W(A) is the numerical range
of A. We show that the conjecture holds in its strong, completely bounded
form, i.e., where p above is allowed to be any matrix-valued polynomial, for
all tridiagonal 3×3 matrices with constant main diagonal: ⎡⎣⎢ac10b1ac20b2a⎤⎦⎥,a,bk,ck∈C, or equivalently, for all complex 3×3 matrices
with elliptic numerical range and one eigenvalue at the center of the ellipse.
We also extend the main result of D. Choi in (Linear Algebra Appl. 438,
3247--3257) slightly.

Last updated on 2020-30-05 at 03:51