CROUZEIX'S CONJECTURE HOLDS FOR TRIDIAGONAL 3 x 3 MATRICES WITH ELLIPTIC NUMERICAL RANGE CENTERED AT AN EIGENVALUE

Crouzeix stated the following conjecture in [Integral Equations Operator Theory, 48 (2004), pp. 461-477]: For every n x n matrix A and every polynomial p, parallel to p(A)parallel to <= 2 max(z is an element of)W(A)vertical bar p(z)vertical bar, 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 x 3 matrices with constant main diagonal, [GRAPHICS] , a, b(k), c(k) is an element of C, or equivalently, for all complex 3 x 3 matrices with elliptic numerical range and one eigenvalue at the center of the ellipse. We also extend the main result of Choi in [Linear Algebra Appl., 438 (2013), pp. 3247-3257] slightly.