Localization and semibounded energy - a weak unique continuation theorem

Let D be a self-adjoint differential operator of Dirac type acting on secti ons in a vector bundle over a closed Riemannian manifold M. Let H be a clos ed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D restricted to H is semibounded. We show that every element psi i s an element of H has the weak unique continuation property, i.e, if psi va nishes on a nonempty open subset of M, then it vanishes on all of M. (C) 20 00 Elsevier Science B.V, All rights reserved.