Gohberg Lemma and Spectral Results for Pseudodifferential Operators on Locally Compact Abelian Groups

2023/06/30 by Jara, Néstor, Măntoiu, Marius
#22C05 #43A70 #47A63 #47G30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2306.17687

Abstract

We provide a new type of proof for known or new Gohberg lemmas for pseudodifferential operators on Abelian locally compact groups X. We use C^*-algebraic techniques, which also give spectral results to which the Gohberg lemma is just a corollary. These results extend most of those appearing in the literature in various directions. In particular, compactness or a a Lie structure are not needed. The ideal of all the compact operators in L²(X) is replaced by all the ideals having a crossed product structure, which is a consistent generalization. We also indicate several new examples, mostly connected to specific behaviors of functions on the dual Ξ of X.

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