Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on \({{\mathbb S}^1}\) (Q611969)

The authors consider the Hörmander classes \(S^m(\mathbb S^1\times \mathbb Z)\) of symbols on the unit circle \(\mathbb S^1\) and study the corresponding pseudodifferential operators. They prove that elliptic pseudodifferential operators on \(\mathbb S^1\) are Fredholm on \(L^p(\mathbb S^1)\), \(1<p<\infty\). These operators have the Fredholm spectral invariance property, that is, the inverse to the image of an operator with a symbol from \(S^0(\mathbb S^1\times \mathbb Z)\) in the Calkin algebra corresponds to a pseudodifferential operator of the same type. This property is used to prove that the ellipticity is a necessary condition for the Fredholmness on \(L^2(\mathbb S^1)\). For operators on \(\mathbb R^n\), such a result was proved by \textit{A. Dasgupta} [Operator Theory: Advances and Applications 189, 107--116 (2009; Zbl 1170.47030)].