We give a new characterization of the class <$Eroman {N sub M sup 0 }> [-1, 1] of the operator-valued in the Hilbert space M Nevanlinna functions that admit representations as compressed resolvents (m-functions) of selfadjoint contractions. We consider the automorphism <$EGAMMA~:~M( lambda )~symbol O~M sub GAMMA ( lambda )~:=~(( lambda sup 2 ~-~1)M( lambda )) sup -1> of the class <$Eroman {N sub M sup 0 }> [-1, 1] and construct a realization of <$E M sub GAMMA ( lambda )> as a compressed resolvent. The unique fixed point of <$EGAMMA> is the m-function of the block-operator Jacobi matrix related to the Chebyshev polynomials of the first kind. We study a transformation <$EGAMMA Hat ~:~M( lambda )~symbol О~M sub { GAMMA Hat } ( lambda )> := <$E-(M( lambda )~+~lambda I sub M ) sup -1> that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point <$EM sub 0> of <$EGAMMA Hat> admits a realization as the compressed resolvent of the "free" discrete Schrodinger operator <$Eroman J Hat sub 0> in the Hilbert space <$Eroman H sub 0 ~=~l sup 2 ({ roman bold N} sub 0 )~symbol д~M>. We prove that <$EM sub 0> is the uniform limit on compact sets of the open upper/lower half-plane in the operator norm topology of the iterations {<$EM sub n+1 ( lambda )~=~-(M sub n ( lambda )~+~ lambda I sub M ) sup -1 }> of <$EGAMMA Hat>. We show that the pair {<$E{ roman bold H sub 0 ,~J Hat sub 0 } }> is the inductive limit of the sequence of realizations {<$Eh Hat sub n ,~A Hat sub n }> of {<$EM sub n }>. In the scalar case (<$EM~=~roman bold C>), applying the algorithm of I. S. Kac, a realization of iterates {<$EM sub n ( lambda )}> as m-functions of canonical (Hamiltonian) systems is constructed.

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