Factorization of operator functions in a Hilbert space (Q1400107)

For a continuous operator function with zero not belonging to the convex hull of its spectrum, it is shown that there is a well-defined notion of the index. If the operator function is Hölder continuous, then it admits a factorization in three special operator function factors, at least two of which are invertible in the closed unit disk. In the general case, for a noncommuting operator family, the condition of non-belonging of zero in the convex hull does not imply the existence of an inner product. A relevant example concerning factorization conditions is given in the last section of the paper.