On Wielandt-Mirsky's conjecture for matrix polynomials

In matrix analysis, the \textit{Wielandt-Mirsky conjecture} states that $$dist(\sigma(A), \sigma(B)) \leq \|A-B\|,$$ for any normal matrices $A, B \in \mathbb C^{n\times n}$ and any operator norm $\|\cdot \|$ on $C^{n\times n}$. Here $dist(\sigma(A), \sigma(B))$ denotes the optimal matching distance between the spectra of the matrices $A$ and $B$. It was proved by A.J. Holbrook (1992) that this conjecture is false in general. However it is true for the Frobenius distance and the Frobenius norm (the Hoffman-Wielandt inequality). The main aim of this paper is to study the Hoffman-Wielandt inequality and some weaker versions of the Wielandt-Mirsky conjecture for matrix polynomials.