Transformations of Nevanlinna operator-functions and their fixed points

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 $\mathbf{\Gamma }:$ $M(\lambda){\mapsto}M_{{\bf \Gamma}}(\lambda):=\left((\lambda^2-1)M(\lambda)\right)^{-1}$oftheclass${\bf N}^0_{\mathfrak M}[-1,1]$andconstructarealizationof$M_{{\bf \Gamma}}(\lambda)$asacompressedresolvent\mathrm{.}Theuniquefixedpointof${\bf\Gamma}$isthe$m$-function of the block-operator Jacobi matrix related to the Chebyshev polynomials of the first kind. We study a transformation $\widehat{\mathbf{\Gamma }}:$ ${\mathcal M}(\lambda)\mapsto {\mathcal M}{{\bf\widehat \Gamma}}(\lambda) :=-({\mathcal M}(\lambda)+\lambda I{\mathfrak M})^{-1}$ that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point ${\mathcal{M}}_0$ of $\widehat{\mathbf{\Gamma }}$ admits a realization as the compressed resolvent of the “free” discrete Schr"{o}dinger operator ${\widehat{\mathbf{J}}}_0$ in the Hilbert space ${\mathbf{H}}_0={\mathrm{\ell }}^2({\mathbb{N}}_0)⨂\mathfrak{M}$. 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${\bf\widehat\Gamma}$\mathrm{.}Weshowthatthepair${{\bf H}_0,{\bf \widehat J}_0}$istheinductivelimitofthesequenceofrealizations${\widehat{\mathfrak H}_n,\widehat A_n}$of${{\mathcal M}_n}$\mathrm{.}Inthescalarcase$({\mathfrak M}={\mathbb C})$,applyingthealgorithmofI\mathrm{.}S\mathrm{.}Kac,arealizationofiterates${{\mathcal M}_n}$as$m$-functions of canonical (Hamiltonian) systems is constructed.