Transformations of Nevanlinna operator-functions and their fixed points

TL;DR

This paper characterizes specific Nevanlinna functions related to selfadjoint contractions, studies their transformations and fixed points, and constructs realizations as compressed resolvents of operators, revealing deep structural properties.

Contribution

It introduces new characterizations of Nevanlinna functions, analyzes their transformations and fixed points, and constructs operator realizations, including in the scalar case using Kac's algorithm.

Findings

Fixed point of automorphism is an m-function of a block-operator Jacobi matrix.

The fixed point of the transformation is realized as a compressed resolvent of a discrete Schrödinger operator.

Iterates of the transformation converge uniformly on compact sets in the operator norm topology.

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 automorphism ${\bf \Gamma}:$ $M(\lambda){\mapsto}M_{{\bf \Gamma}}(\lambda):=\left((\lambda^2-1)M(\lambda)\right)^{-1}$ 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 Schr\"{o}dinger 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 $\{{\mathcal M}_n\}$ as $m$-functions of canonical (Hamiltonian) systems is constructed.