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 \(\textsf{X}\) and an important role is played by its Pontryagin dual \({\widehat{\textsf{X}}}\) . Hörmander-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 \(\ell ^\infty ({\widehat{\textsf{X}}})\) .
