Perturbation of Sectorial Projections of Elliptic Pseudo-differential Operators

TL;DR

This paper studies how sectorial projections of elliptic pseudo-differential operators depend continuously on the operators themselves, with applications to boundary value problems and an analysis of topological obstructions.

Contribution

It establishes the continuity of sectorial projections in a specific topology and applies this to curves of elliptic operators, including boundary value problems, while identifying topological obstructions.

Findings

Sectorial projections depend continuously on elliptic operators.

Continuity of Calderon projections and Cauchy data spaces under certain conditions.

Identification of topological obstructions in existing proofs.

Abstract

Over a closed manifold, we consider the sectorial projection of an elliptic 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 operators is equipped with a certain topology which we explicitly describe. Our main application deals with a continuous curve of arbitrary first order linear elliptic differential operators over a compact manifold with boundary. Under the additional assumption of the weak inner unique continuation property, we derive the continuity of a related curve of Calderon projections and hence of the Cauchy data spaces of the original operator curve. In the Appendix, we describe a topological obstruction against a verbatim use of R. Seeley's original argument for the complex powers, which was seemingly overlooked in previous studies of the sectorial projection.