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In this paper we prove a version of the Gohberg lemma on compact Lie groupsgiving an estimate from below for the distance from a given operator to the setof compact operators on compact Lie groups. As a consequence, we prove severalresults on bounds for the essential spectrum and a criterion for an operator tobe compact. The conditions are given in terms of the matrix-valued symbols ofoperators.

Gohberg lemma, compactness, and essential spectrum of operators on compact Lie groups