Elliptic operators with infinite-dimensional state spaces

Abstract.

Motivated by applications to problems from physics, we study elliptic operators with operator-valued coefficients acting on Banach-space-valued distributions. After giving a definition of ellipticity, normal ellipticity in particular, generalizing the classical concepts, we show that normally elliptic operators are negative generators of analytic semigroups on $L_p({\Bbb R}^n, E)$ for 1 $\leq p < \infty$ and on $BUC({\Bbb R}^n, E)$ and $C_0({\Bbb R}^n, E)$, as well as on all Besov spaces of E-valued distributions on ${\Bbb R}^n$, where E is any Banach space. This is true under minimal regularity assumptions for the coefficients, thanks to a point-wise multiplier theorem for E-valued distributions proven in the appendix.