Generalized Nevanlinna functions and multivalency

A new characterization of generalized Nevanlinna functions in terms of multivalency is established. By analyzing multivalent functions additional novel characterizations of (generalized) Nevanlinna functions are obtained. As a particular consequence of these investigations a function-theoretic proof of the factorization property of generalized Nevanlinna functions is obtained.     

The factorization of generalized Nevanlinna functions and the invariant subspace property

The well-known invariant subspace property of selfadjoint relations (multi-valued operators) in Pontryagin spaces is shown to be equivalent to the factorization property of (scalar) generalized Nevanlinna functions. This connection is established by a new realization for generalized Nevanlinna functions explicitly reflecting this connection. Combining this result with the new function-theoretic proof for the factorization property of generalized Nevanlinna functions contained in Wietsma (2018) immediately yields a new proof for the invariant subspace property of selfadjoint relations in Pontryagin spaces.     

Representations of Unitary Relations Between Kreĭn Spaces

The structure of unitary relations between Kreĭn spaces is investigated in geometrical terms. Two approaches are presented: The first approach relies on the so-called Weyl identity and the second approach is based on a graph decomposition of unitary relations. As a consequence of these investigations a quasi-block and a proper block representation of unitary operators are established. Both approaches yield also several new necessary and sufficient conditions for isometric relations to be unitary.