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Classically, Gohberg-type Lemmas provide lower bounds for the distance ofsuitable pseudodifferential operators acting in a Hilbert space to the ideal ofcompact operators, in terms of "the behavior of the symbol at infinity". Inthis article the pseudodifferential operators are associated to a compactAbelian 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 vanishingoscillation condition, which anyway is more general even in the particularcases allowing a full pseudodifferential calculus. In addition, the distance toa large class of operator ideals is controlled; the compact operators only forma particular case. This involves invariant closed subsets of certaincompactifications of the dual group or, equivalently, invariant ideals of$\ell^\infty(\widehat X)$.

Anisotropic Gohberg Lemmas for Pseudodifferential Operators on Abelian Compact Groups