The factorization of generalized Nevanlinna functions and the invariant subspace property

Abstract

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.