Exponential bounds for noncommuting systems of matrices (Q2717556)

Let \(T=(T_1,\dots,T_d)\) be a system of bounded linear operators acting in a Banach space and \(\langle T, \zeta \rangle=\sum_{j=1}^{d} T_j \zeta_j; \zeta \in C^d\). It is well known that the bound \(\|e^{i\langle T, \zeta \rangle}\|\leq C_0 (1+|\zeta |)^s e^{r |Im \zeta |} \forall \zeta \in C^d\) implies that \(\sigma (\langle T, \xi \rangle) \subset R \forall \xi \in R^d\). The aim of the paper is to prove a somewhat converse result: if \(T=(T_1,\dots,T_d)\) is a \(d\)-tuple of \(n \times n\) matrices such that \(\sigma (\langle T, \xi \rangle) \subset R \forall \xi \in R^d\) then there exist \(C_0>0\) and \(r \geq 0\) such that \(\|e^{i \langle T, \zeta \rangle}\|\leq (1+|\zeta |)^{n-1} e^{r |Im \zeta|} \forall \zeta \in C^d\). The proof appeals to the monogenic functional calculus.