We define a spectrum for Lipschitz continuous nonlinear operators in Banach spaces by means of a certain kind of "pseudo-adjoint" and study some of its properties.

It is shown that a finite system T of matrices whose real linear combinations have real spectrum satisfies a bound of the form $\left|\right|e^{i⟨T,\zeta ⟩}\left|\right|\le C(1+{\left|\zeta \right|)}^s{e}^{r\left|\Im \zeta \right|}$. The proof appeals to the monogenic functional calculus.