Trace conditions for regular spectral behavior of vector-valued analytic functions

A new class of Banach algebra valued functions is identified for which the logarithmic residue with respect to a Cauchy domain ? vanishes (if and) only if the functions take invertible values in ?. Trace conditions and the extraction of elementary factors of the type e - p + (? - ?) 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.