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New classes of generalized Nevanlinna functions, which under multiplicationwith an arbitrary fixed symmetric rational function remain generalizedNevanlinna functions, are introduced. Characterizations for these classes offunctions are established by connecting the canonical factorizations of theproduct function and the original generalized Nevanlinna function in aconstructive manner. Also a detailed functional analytic treatment of theseclasses of functions is carried out by investigating the connection between therealizations of the product function and the original function. The operatortheoretic treatment of these realizations is based on the notions of riggedspaces, boundary triplets, and associated Weyl functions.

Products of generalized Nevanlinna functions with symmetric rational functions