Parameter-dependent Pseudodifferential Operators of Toeplitz Type

TL;DR

This paper develops a calculus for parameter-dependent pseudodifferential operators, including Toeplitz type, characterizing ellipticity via principal symbols, and constructing parametrices for large parameters, with applications to invertibility of specific operator forms.

Contribution

It introduces a new calculus for parameter-dependent pseudodifferential operators of Toeplitz type, including novel symbol structures and ellipticity criteria.

Findings

Characterization of parameter-ellipticity via principal symbols.

Construction of parametrices for large parameters.

Analysis of invertibility for Toeplitz type operators with projections.

Abstract

We present a calculus of pseudodifferential operators that contains both usual parameter-dependent operators -- where a real parameter \tau\ enters as an additional covariable -- as well as operators not depending on \tau. Parameter-ellipticity is characterized by the invertibility of three associated principal symbols. The homogeneous principal symbol is not smooth on the whole co-sphere bundle but only admits directional limits at the north-poles, encoded by a principal angular symbol. Furthermore there is a limit-family for \tau\to+\infty. Ellipticity permits to construct parametrices that are inverses for large values of the parameter. We then obtain sub-calculi of Toeplitz type with a corresponding symbol structure. In particular, we discuss invertibility of operators of the form P_1A(\tau)P_0 where both P_0 and P_1 are zero-order projections and A(\tau) is a usual parameter-dependent operator of arbitrary order or A(\tau)=\tau^\mu-A with a pseudodifferential operator A of positive integer order \mu.