Crouzeix’s conjecture and perturbed Jordan blocks

Crouzeix conjectured in [M. Crouzeix, Bounds for analytical functions of matrices, Integr. Equ. Oper. Theory 48 (2004) 461–477] that for any square matrix A and any polynomial p,‖p(A)‖⩽2maxp(z)|:z∈W(A)},‖p(A)‖⩽2maxp(z)|:z∈W(A)},where W(A)W(A) is the field of values of A   and ‖·‖‖·‖ denotes the spectral norm. In this paper, we show that the conjecture holds for matrices of the formλ1⋱⋱⋱1νλ,where λλ and νν are complex numbers. The technique, for |ν|≪1|ν|≪1 and for |ν|≫1|ν|≫1, is to show that if g is a bijective conformal mapping from the field of values of such a matrix A   to the unit disk, mapping λλ to 0, then g(A)g(A) is similar to a contraction via a similarity transformation with condition number at most 2.