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Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on \({{\mathbb S}^1}\)

  • Shahla Molahajloo
  • M. W. Wong
Article

Abstract

Analogues of pseudo-differential operators of the Hörmander class for the unit circle \({{\mathbb S}^1}\) centered at the origin are studied. We prove that elliptic pseudo-differential operators on \({{\mathbb S}^1}\) are Fredholm on \({L^p({\mathbb S}^1),1 < p < \infty.}\) Then we prove the spectral invariance of these operators in \({L^2({\mathbb S}^1)}\) and we use the spectral invariance to prove that ellipticity is a necessary condition for Fredholmness on \({L^2({\mathbb S}^1)}\).

Keywords

Pseudo-differential operators Sobolev spaces Fredholmness Ellipticity Spectral invariance Calkin algebra 

Mathematics Subject Classification (2000)

47A53 47G30 

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© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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