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

Abstract: M. Crouzeix formulated the following conjecture in (Integral Equations Operator Theory 48, 2004, 461--477): For every square matrix $A$ and every polynomial $p$, $$ |p(A)| \le 2 \max_{z\in W(A)}|p(z)|, $$ 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 $3\times 3$ matrices with constant main diagonal: $$ \left[\begin{matrix}a&b_1&0\c_1&a&b_2\0&c_2&a\end{matrix}\right],\qquad a,b_k,c_k\in\mathbb C, $$ or equivalently, for all complex $3\times 3$ 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.