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It is shown that the so-called principal function invariant, which is associated in a unitarily invariant way to operators with trace class self commutator TT* -T*T, is invariant under trace class perturbations of T and is an extension of the index of T-z to the whole plane. The connection of the principal function, under additional hypothesis, with the determination of the maximal ideal space of the C* algebra generated by T is discussed, and it is shown that the principal function, even whendoi:10.1073/pnas.71.5.1952 pmid:16592156 pmcid:PMC388361 fatcat:trzavpybojfijnal6nuwajfmw4