On the Fredholm property of bisingular pseudodifferential operators

TL;DR

This paper establishes the equivalence between ellipticity and the Fredholm property for bisingular pseudodifferential operators on Euclidean spaces and manifolds, also proving spectral invariance and extending results to Toeplitz operators.

Contribution

It proves the equivalence of ellipticity and Fredholm property for bisingular operators on manifolds and Euclidean spaces, and extends spectral invariance to Toeplitz-type operators.

Findings

Ellipticity is equivalent to the Fredholm property for these operators.

Spectral invariance holds for the classes studied.

Results extend to Toeplitz type operators.

Abstract

For operators belonging either to a class of global bisingular pseudodifferential operators on $R^m \times R^n$ or to a class of bisingular pseudodifferential operators on a product $M \times N$ of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain associated homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to the even larger classes of Toeplitz type operators.