Rudi Wietsma: The product factorization of generalized Nevanlinna functions

A fascinating characterization of (scalar) generalized Nevanlinna functions is that they can be factorized as the product of a symmetric rational function and an ordinary (scalar) Nevanlinna function. This characterization of generalized Nevanlinna functions gives easily insight into their essential properties. Moreover, this factorization can be used to understand the spectral behavior of self-adjoint operators in Pontryagin spaces (a generalization of Hilbert spaces). Also in the case of operator-valued generalized Nevanlinna functions a similar factorization has been established; however, there are still unanswered questions remaining about it.

During this lecture an overview is given of the available proofs for the afore-mentioned factorization property in the scalar case, whereby special attention is given to the interplay between function-theoretical proofs and operator-theoretical proofs. Moreover, a completely new proof for the factorization property (in the scalar case) is presented which is based on new characterizations of generalized Nevanlinna functions. Thereby hoping to open also a path for the better understanding of the factorization property in the operator-valued case.