On the spectral flow theorem of Robbin-Salamon for finite intervals

In this article we consider operators of the form ${\mathrm{\partial }}_s\xi +A(s)\xi$ where $s$ lies in an interval $[-T,T]$ and $s\mapsto A(s)$ is continuous. Without boundary conditions these operators are not Fredholm. However, using interpolation theory one can define suitable boundary conditions for these operators so that they become Fredholm. We show that in this case the Fredholm index is given by the spectral flow of the operator path $A$.