Elliptic operators with infinite-dimensional state spaces

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 for 1 and on and , as well as on all Besov spaces of E-valued distributions on , 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. Received August 23, 2000; accepted December 12, 2000.