A proof of Crouzeix's conjecture for a class of matrices

Crouzeix's conjecture is that for any square matrix A and any polynomial p we have||p(A)||⩽2maxp(z)|:z∈W(A)},where W(A) is the field of values of A and ||·|| denotes the spectral norm. In this paper, we show that the conjecture holds for the matrices of the formλα1⋱⋱⋱αn-1αnλ,where λ∈C and α=(α1,…,αn)∈Cn.