Ellipticity in Pseudodifferential Algebras of Toeplitz Type

TL;DR

This paper investigates how ellipticity properties in a broad algebra of pseudodifferential operators transfer to a subalgebra formed by projections, with applications to boundary value problems and singularity analysis.

Contribution

It provides a detailed analysis of ellipticity descent, parametrix construction, and Fredholm properties in subalgebras of pseudodifferential operators, extending to parameter-dependent cases.

Findings

Ellipticity in the larger algebra descends to the subalgebra under certain conditions.

Constructs parametrices and establishes Fredholm properties for the subalgebra.

Demonstrates invertibility of the Stokes operator with Dirichlet boundary conditions.

Abstract

Let L^\star be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T^\star be a subalgebra of operators of the form P_1AP_0, where P_0 and P_1 are two projections. The elements of L^\star act as linear continuous operators in certain scales of abstract Sobolev spaces, the elements of the subalgebra in the corresponding subspaces determined by the projections. We study how the ellipticity in L^\star descends to T^\star, focusing on parametrix construction, Fredholm property, and homogeneous principal symbols. Applications concern SG-pseudodifferential operators, pseudodifferential operators on manifolds with conical singularities, and Boutet de Monvel's algebra for boundary value problems. In particular, we derive invertibilty of the Stokes operator with Dirichlet boundary conditions in a subalgebra of Boutet de Monvel's algebra. We indicate how the concept generalizes to parameter-dependent operators.