Transformations of Nevanlinna operator-functions and their fixed points

Abstract

We give a new characterization of the class ${\mathbf{N}}_{\mathfrak{M}}^0[-1,1]$ 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 $\mathbf{\Gamma }:$ $M(\lambda )\mapsto M_{\mathbf{\Gamma }}(\lambda ):={\left(({\lambda }^2-1)M(\lambda )\right)}^{-1}$ of the class ${\mathbf{N}}_{\mathfrak{M}}^0[-1,1]$ and construct a realization of $M_{\mathbf{\Gamma }}(\lambda )$ as a compressed resolvent. The unique fixed point of $\mathbf{\Gamma }$ is the $m$-function of the block-operator Jacobi matrix related to the Chebyshev polynomials of the first kind. We study a transformation $\hat{\mathbf{\Gamma }}:$ $\mathcal{M}(\lambda )\mapsto {\mathcal{M}}_{\hat{\mathbf{\Gamma }}}(\lambda ):=-(\mathcal{M}(\lambda )+\lambda I_{\mathfrak{M}}{)}^{-1}$ that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point ${\mathcal{M}}_0$ of $\hat{\mathbf{\Gamma }}$ admits a realization as the compressed resolvent of the "free" discrete Schrödinger operator ${\hat{\mathbf{J}}}_0$ in the Hilbert space ${\mathbf{H}}_0={\ell }^2({\mathbb{N}}_0)⨂\mathfrak{M}$. 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 $\hat{\mathbf{\Gamma }}$. We show that the pair $\{{\mathbf{H}}_0,{\hat{\mathbf{J}}}_0\}$ is the inductive limit of the sequence of realizations $\{{\hat{\mathfrak{H}}}_n,{\hat{A}}_n\}$ 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.

Accepted for publication in the Methods of Functional Analysis and Topology