Crouzeix's conjecture holds for a subclass of tridiagonal 3x3 matrices

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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 for tridiagonal
3×3 matrices ⎡⎣⎢ac0bac0ba⎤⎦⎥,a,b,c∈C, with constant diagonals, even if any
number of off-diagonal elements should be set to zero. We also extend the main
result of D. Choi in (Linear Algebra Appl. 438, 3247--3257) slightly.

Last updated on 2020-03-07 at 03:35