Realization of nonstrict matrix Nevanlinna functions as Weyl functions of symmetric operators in Pontryagin spaces

Abstract

Matrix-valued Nevanlinna functions with possibly noninvertible imaginary part are realized as Q-functions or Weyl functions of symmetric operators in Pontryagin spaces. The functions are decomposed into a constant part, which gives rise to a realization in a finite dimensional Pontryagin space K, and a strict or uniformly strict part, which gives rise to a realization in a Hilbert space H. A coupling procedure then leads to a symmetric operator in the product space H x K and to the realization of the given Nevanlinna function.