Let f be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivative f′f-1 around a Cauchy domain D vanishes. Does it follow that f takes invertible values on all of D? 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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doi.org/10.1007/BF01206410, hdl.handle.net/1765/71775
Integral Equations and Operator Theory
Erasmus School of Economics

Bart, H., Ehrhardt, T., & Silbermann, B. (1994). Logarithmic residues in Banach algebras. Integral Equations and Operator Theory, 19(2), 135–152. doi:10.1007/BF01206410