M. Crouzeix formulated the following conjecture in (Integral EquationsOperator Theory 48, 2004, 461--477): For every square matrix A and everypolynomial p,∥p(A)∥≤2maxz∈W(A)|p(z)|, where W(A) is thenumerical range of A. We show that the conjecture holds for tridiagonal3×3 matrices ⎡⎣⎢ac0bac0ba⎤⎦⎥,a,b,c∈C, with constant diagonals, even if anynumber of off-diagonal elements should be set to zero. We also extend the mainresult of D. Choi in (Linear Algebra Appl. 438, 3247--3257) slightly.
|Publication status||Published - 2017|
|MoE publication type||D4 Published development or research report or study|