A periodic Faddeev-type solution operator

We construct periodic solution operators for the equation Delta u + 2i zeta . del u = f in a bounded domain with the help of Fourier series. We prove that the L(2)-norms of these operators converge to zero if the parameter \Im zeta\ goes to infinity. Then we apply these operators to show that functions u epsilon C-0(2)(R(d)) satisfying an inequality \Delta u(x)\ less than or equal to M \u(x)\ in R(d) must vanish everywhere. We extend this result to other second order elliptic differential operators with constant coefficients replacing the Laplacian. Finally, we use the solution operators to derive that the span of products of solutions to differential equations is dense in L(1). (C) 1996 Academic Press, Inc.