On Two Problems In The Theory Of Elliptic Operators On Manifolds

In this thesis, we study two problems in the theory of elliptic operators:The first one is how the continuity of a curve of sectorial projections can be derivedwithin the symbolic calculus, supplemented by estimates of some smoothing operators.More precisely, over a closed manifold, we consider the sectorial projection of an el-liptic pseudo-differential operator A of positive order with two rays of minimal growth.We show that it depends continuously on A when the space of pseudo-differential op-erators is equipped with a certain topology which we explicitly describe. Our main ap-plication deals with a continuous curve of arbitrary first order linear elliptic differentialoperators over a compact manifold with boundary. Under the additional assumption ofthe weak inner unique continuation property, we derive the continuity of a related curveof Caldero′n projections and hence of the Cauchy data spaces of the original operatorcurve.The second problem is to prove a generalized version of ‘Fredholm index=spec-tral flow’ theorem on infinite cylinders. More precisely, let E be a Hermitian bundleover a closed manifold, and {A(t)}t∈Rbe a family of first order elliptic differential op-erators (non-self-adjoint in general) on E. We prove that under some natural conditions,d/dt A(t) is Fredholm, and its index equals to spectral flow of {A(t)}t∈R.