Transformations of Nevanlinna operator-functions and their fixed points
We give a new characterization of the class N^0_M[-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 Γ: M(λ)M_Γ (λ):=((λ^2-1)M(λ))^-1 of the class N^0_M[-1,1] and construct a realization of M_Γ (λ) as a compressed resolvent. The unique fixed point of Γ is the m-function of the block-operator Jacobi matrix related to the Chebyshev polynomials of the first kind. We study a transformation Γ: M(λ)M_Γ (λ) :=-(M(λ)+λ I_M)^-1 that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point M_0 of Γ admits a realization as the compressed resolvent of the "free" discrete Schrödinger operator J_0 in the Hilbert space H_0=^2(N_0)M. We prove that M_0 is the uniform limit on compact sets of the open upper/lower half-plane in the operator norm topology of the iterations {M_n+1(λ)=-(M_n(λ)+λ I_M)^-1\