  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
$\ell^\infty(\widehat X)$.
