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Parameter-dependent pseudodifferential operators of Toeplitz type

Research paper by Jörg Seiler

Indexed on: 08 Aug '13Published on: 08 Aug '13Published in: Annali di Matematica Pura ed Applicata (1923 -)



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 cosphere bundle but only admits directional limits at the north-poles, encoded by a principal angular symbol. Furthermore, there is a limit-family for \(\tau \rightarrow +\infty \). Ellipticity permits to construct parametrices that are inverses for large values of the parameter. We then obtain subcalculi 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 \(\mu \).