Vector valued logarithmic residues and the extraction of elementary factors

An analysis is presented of the circumstances under which, by the extraction of elementary factors, an analytic Banach algebra valued function can be transformed into one taking invertible values only. Elementary factors are generalizations of the simple scalar expressions λ – α, the building blocks of scalar polynomials. In the Banach algebra situation they have the form e – p + (λ – α)p with p an idempotent. The analysis elucidates old results (such as on Fredholm operator valued functions) and yields new insights which are brought to bear on the study of vector-valued logarithmic residues. These are contour integrals of logarithmic derivatives of analytic Banach algebra valued functions. Examples illustrate the subject matter and show that new ground is covered. Also a long standing open problem is discussed from a fresh angle.