 
Summary: Elliptic Operators with InniteDimensional State Spaces
Herbert Amann
Abstract. Motivated by applications to problems from physics, we study elliptic operators with
operatorvalued coeÆcients acting on Banachspacevalued distributions. After giving a deni
tion of ellipticity, normal ellipticity in particular, generalizing the classical concepts, we show
that normally elliptic operators are negative generators of analytic semigroups on Lp (R n ; E) for
1 p < 1, and on BUC(R n ; E) and C 0 (R n ; E), as well as on all Besov spaces of Evalued distri
butions on R n , where E is any Banach space. This is true under minimal regularity assumptions
for the coeÆcients, thanks to a pointwise multiplier theorem for Evalued distributions proven
in the appendix.
Mathematical Subject Classication (1991). 35J30, 35J45, 35K25, 35S50, 46F25, 47D06,
47G99 .
Keywords. Elliptic operators with operatorvalued coeÆcients, resolvent estimates, analytic
semigroups, vectorvalued Besov spaces, Lebesgue spaces, and spaces of continuous and Holder
continuous functions, pointwise multipliers for vectorvalued Besov spaces.
Introduction
In this paper we derive resolvent estimates for linear elliptic dierential operators
A := A(x; D) :=
X
jjm
