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Transformations of Nevanlinna operator-functions and their fixed points


Abstract

We give a new characterization of the class ${\bf N}^0_{\mathfrak M}[-1,1]$ of the operator-valued in the Hilbert space ${\mathfrak M}$ Nevanlinna functions that admit representations as compressed resolvents ($m$-functions) of selfadjoint contractions. We consider the auto\-morphism ${\bf \Gamma}:$ $M(\lambda){\mapsto}M_{{\bf \Gamma}} (\lambda):=\left((\lambda^2-1)M(\lambda)\right)^{-1}$ of the class ${\bf N}^0_{\mathfrak M}[-1,1]$ and construct a realization of $M_{{\bf \Gamma}}(\lambda)$ as a compressed resolvent. The unique fixed point of ${\bf\Gamma}$ is the $m$-function of the block-operator Jacobi matrix related to the Chebyshev polynomials of the first kind. We study a transformation ${\bf\widehat \Gamma}:$ ${\mathcal M}(\lambda)\mapsto {\mathcal M}_{{\bf\widehat \Gamma}}(\lambda) :=-({\mathcal M}(\lambda)+\lambda I_{\mathfrak M})^{-1}$ that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point ${\mathcal M}_0$ of ${\bf\widehat \Gamma}$ admits a realization as the compressed resolvent of the "free" discrete Schrödinger operator ${\bf\widehat J}_0$ in the Hilbert space ${\bf H}_0=\ell^2({\mathbb N}_0)\bigotimes{\mathfrak M}$. We prove that ${\mathcal M}_0$ is the uniform limit on compact sets of the open upper/lower half-plane in the operator norm topology of the iterations $\{{\mathcal M}_{n+1}(\lambda)=-({\mathcal M}_n(\lambda)+\lambda I_{\mathfrak M})^{-1}\}$ of ${\bf\widehat\Gamma}$. We show that the pair $\{{\bf H}_0,{\bf \widehat J}_0\}$ is the inductive limit of the sequence of realizations $\{\widehat{\mathfrak H}_n,\widehat A_n\} $ of $\{{\mathcal M}_n\}$. In the scalar case $({\mathfrak M}={\mathbb C})$, applying the algorithm of I.S. Kac, a realization of iterates $\{{\mathcal M}_n(\lambda)\}$ as $m$-functions of canonical (Hamiltonian) systems is constructed.

Key words: Nevanlinna operator-valued function, compressed resolvent, fixed point, block-operator Jacobi matrix, canonical system.


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Article Information

TitleTransformations of Nevanlinna operator-functions and their fixed points
SourceMethods Funct. Anal. Topology, Vol. 23 (2017), no. 3, 212-230
MilestonesReceived 19/04/2017; Revised 30/05/2017
CopyrightThe Author(s) 2017 (CC BY-SA)

Authors Information

Yu. M. Arlinskiĭ
Department of Mathematics, Dragomanov National Pedagogical University, 9 Pirogova, Kyiv, 01601, Ukraine


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Citation Example

Yu. M. Arlinskiĭ, Transformations of Nevanlinna operator-functions and their fixed points, Methods Funct. Anal. Topology 23 (2017), no. 3, 212-230.


BibTex

@article {MFAT984,
    AUTHOR = {Arlinskiĭ, Yu. M.},
     TITLE = {Transformations of Nevanlinna operator-functions and their fixed points},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {23},
      YEAR = {2017},
    NUMBER = {3},
     PAGES = {212-230},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=984},
}


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