Perturbation of sectorial projections of elliptic pseudo-differential operators. (English) Zbl 1257.58017
Summary: 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 Calderón projections and hence of the Cauchy data spaces of the original operator curve. In the Appendix, we describe a topological obstruction to a verbatim use of R. Seeley’s original argument for the complex powers, which was seemingly overlooked in previous studies of the sectorial projection.
MSC:
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 58J37 | Perturbations of PDEs on manifolds; asymptotics |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 58J05 | Elliptic equations on manifolds, general theory |
Keywords:
sectorial projections; elliptic operators; pseudo-differential operators; non-symmetric operators; Calderón projection; Cauchy data spacesReferences:
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