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.
Categories: math.FA