Commutation properties of the coefficient matrix in the differential equation of an inner function

Abstract:

Let $A(x)$ be an operator valued function that is analytic on the real axis. Assume that $A(x)$ is selfadjoint for each real x. It is shown that $A(x)$ and $\smallint _0^xA(s)$ ds commute for all real x iff $A(x)$ and $A(y)$ commute for all real x and y. This result is then used to establish several new characterizations of the Potapov inner functions of normal operators T such that $\left \| T \right \| < 1$. The case where $\left \| T \right \| = 1, r(T) < 1$ and ${A_T}(x)$ and ${A_T}(y)$ commute for real x and y is discussed. Here ${A_T}(x) = - i{U’_T}(x){U_T}{(x)^\ast }$ and ${U_T}(x)$ is the Potapov inner function for T.