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Holomorphic Operator-valued Functions Generated by Passive Selfadjoint Systems

  • Yuri ArlinskiĭEmail author
  • Seppo Hassi
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 272)

Abstract

Let M be a Hilbert space. In this paper we study a class \(\mathcal{RS}{\mathfrak(m)}\) of operator functions that are holomorphic in the domain \(\mathbb{C} \setminus \{(-\infty, -1] \ \cup \ [1, +\infty)\}\) and whose values are bounded linear operators in \(\mathfrak{m}\). The functions in \(\mathcal{RS}{\mathfrak(m)}\) are Schur functions in the open unit disk \(\mathbb{D}\) and, in addition, Nevanlinna functions in \(\mathbb{C}_{+} \cup \mathbb{C}_{-}\). Such functions can be realized as transfer functions of minimal passive selfadjoint discrete-time systems.We give various characterizations for the class \(\mathcal{RS}{\mathfrak(m)}\) and obtain an explicit form for the inner functions from the class \(\mathcal{RS}{\mathfrak(m)}\) as well as an inner dilation for any function from \(\mathcal{RS}{\mathfrak(m)}\). We also consider various transformations of the class \(\mathcal{RS}{\mathfrak(m)}\), construct realizations of their images, and find corresponding fixed points.

Keywords

Passive system transfer function Nevanlinna function Schur function fixed point 

Mathematics Subject Classification (2010)

Primary 47A48 93B28 93C25 Secondary 47A56 93B20 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Volodymyr Dahl East Ukrainian National UniversitySeverodonetskUkraine
  2. 2.Department of Mathematics and StatisticsUniversity of VaasaVaasaFinland

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