Anisotropic Gohberg Lemmas for Pseudodifferential Operators on Abelian Compact Groups

TL;DR

This paper extends Gohberg-type lemmas to pseudodifferential operators on compact Abelian groups using crossed product C*-algebras, providing bounds on their distance to various operator ideals based on symbol behavior at infinity.

Contribution

It introduces a novel framework for Gohberg lemmas on Abelian groups using crossed product C*-algebras and vanishing oscillation conditions, broadening applicability beyond classical symbol classes.

Findings

Established lower bounds for operator distances in new algebraic settings.

Extended the theory to include a wide class of operator ideals beyond compact operators.

Provided a general approach applicable to various Abelian group structures.

Abstract

Classically, Gohberg-type Lemmas provide lower bounds for the distance of suitable pseudodifferential operators acting in a Hilbert space to the ideal of compact operators, in terms of "the behavior of the symbol at infinity". In this article the pseudodifferential operators are associated to a compact Abelian group $X$ and an important role is played by its Pontryagin dual $\widehat X$. H\"ormander-type classes of symbols are not always available; they will be replaced by crossed product $C^*$-algebras involving a vanishing oscillation condition, which anyway is more general even in the particular cases allowing a full pseudodifferential calculus. In addition, the distance to a large class of operator ideals is controlled; the compact operators only form a particular case. This involves invariant closed subsets of certain compactifications of the dual group or, equivalently, invariant ideals of $\ell^\infty(\widehat X)$.