A note on factorization of vector valued functions involving the extraction of elementary factors

Abstract

This note contains reflections on a type of factorization that came up in the work of analytic operator valued functions by I.C. Gohberg and E.I. Sigal (cf. [14], [10], [11]). The determining characteristic of the factorizations in question is that they are obtained by the extraction of elementary factors of the form I−P+λP where P is an idempotent in the underlying Banach algebra. It will be demonstrated here that there is a fundamental difference between the finite dimensional (matrix) case and the situation where infinite dimensional Banach algebras are taken into consideration.