Abstract elliptic operator and its associated semigroup in a locally convex space (Q1322530)

The author constructs the solution operator for the equation (1) \(Lu= -f\), where \(L\) is an abstract second-order elliptic differential operator defined in a locally compact Hausdorff space \(X\). By giving a locally convex topology to \(C_b(X)\), the space of bounded continuous functions, he proves that the solution operator of (1) is an extension of a classical Green operator [see \textit{M. Itô}, Nat. Sci. Rep. Ochanomizu Univ. 34, 1-13 (1983; Zbl 0533.31006)]. He constructs also the semigroup associated to \(L\) in \(C_b(X)\) equipped with the above-mentioned topology. Finally, an application to a second-order elliptic operator is given.