Rules for the partial multiplicities of the product of holomorphic matrix functions

In 4] I. Gohberg and M.A. Kaashoek formulated the question how the partial multiplicities of the product AB of two holomorphic matrix (or operator) functions are related to those of the factors A and B. E.I. Sigal presented a set of necessary relations in 7], and recently L. Rodman and M. Schaps (6]) obtained another set of relations which must hold. Here we prove a generalization of Sigal's rules which form a proper superset of the rules obtained by Sigal, Rodman and Schaps. Unlike Rodman and Schaps we formulate all results for matrix functions of fixed nxn-size; this leads to another set of relations which are derived for fixed n at first; however, we prove that such rules must hold independently of n, once they are derived for fixed n. Finally we show that this new set of relations completely determines the case of 3×3-matrices.