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

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Abstract

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.

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References

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  1. Abteilung Mathematik, Universität Dortmund, Postfach 500 500, D-4600, Dortmund 50, Germany

    G. Philip A. Thijsse

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  1. G. Philip A. Thijsse
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Thijsse, G.P.A. Rules for the partial multiplicities of the product of holomorphic matrix functions. Integr equ oper theory 3, 515–528 (1980). https://doi.org/10.1007/BF01702314

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