A new class of Banach algebra valued functions is identified for which the logarithmic residue with respect to a Cauchy domain Δ vanishes (if and) only if the functions take invertible values in Δ. Trace conditions and the extraction of elementary factors of the type e - p + (λ - α) p play an important role. The class contains the Fredholm operator valued functions and the Banach algebra valued functions possessing a simply meromorphic resolvent as special instances. An example is given to show that new ground is covered and a long standing open problem is discussed from a fresh angle.

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doi.org/10.1016/j.laa.2008.11.004, hdl.handle.net/1765/18273
Linear Algebra and Its Applications
Erasmus School of Economics

Bart, H., Ehrhardt, T., & Silbermann, B. (2009). Trace conditions for regular spectral behavior of vector-valued analytic functions. Linear Algebra and Its Applications, 430(8-9), 1945–1965. doi:10.1016/j.laa.2008.11.004