Abstract
Letf be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivativef′f −1 around a Cauchy domainD vanishes. Does it follow thatf takes invertible values on all ofD? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues.
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Bart, H., Ehrhardt, T. & Silbermann, B. Logarithmic residues in Banach algebras. Integr equ oper theory 19, 135–152 (1994). https://doi.org/10.1007/BF01206410
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DOI: https://doi.org/10.1007/BF01206410