Semi-Bounded Restrictions of Dirac Type Operators and the Unique Continuation Property

Abstract

Let M be a connected Riemannian manifold and let D be a Dirac type operator acting on smooth compactly supported sections in a Hermitian vector bundle over M. Suppose D has a self-adjoint extension A in the Hilbert space of square-integrable sections. We show that any $L^2$-section $ϕ$ contained in a closed A-invariant subspace onto which the restriction of A is semi-bounded has the unique continuation property: if $ϕ$ vanishes on a non-empty open subset of M, then it vanishes on all of M.