2017/06/03 by Yu. M. Arlinskiĭ, Arlinskiĭ, Yu. M.
Computer Science · Mathematics · #47A06 #47A56 #47B25 #47B36 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1706.00982
openalex publication_date 2017/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a new characterization of the class \bf N0\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 Γ: M(λ)↦M\bf Γ(λ):=((λ2-1)M(λ))-1 of the class \bf N0\mathfrak M[-1,1] and construct a realization of M\bf Γ(λ) as a compressed resolvent. The unique fixed point of \bfΓ 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 Γ: \mathcal M(λ)↦ \mathcal M\bf\widehat Γ(λ) :=-(\mathcal M(λ)+λI\mathfrak M)-1 that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point \mathcal M0 of \bf\widehatΓ admits a realization as the compressed resolvent of the "free" discrete Schrödinger operator \bf\widehat J0 in the Hilbert space \bf H0=ℓ2(\mathbb N0)\bigotimes\mathfrak M. We prove that \mathcal M0 is the uniform limit on compact sets of the open upper/lower half-plane in the operator norm topology of the iterations \\mathcal Mn+1(λ)=-(\mathcal Mn(λ)+λI_\mathfrak M)-1\ of \bf\widehatΓ. We show that the pair \\bf H0,\bf \widehat J0\ is the inductive limit of the sequence of realizations \\widehat\mathfrak Hn,\widehat An\ of \\mathcal Mn\. In the scalar case (\mathfrak M=\mathbb C), applying the algorithm of I.S.~Kac, a realization of iterates \\mathcal Mn\ as m-functions of canonical (Hamiltonian) systems is constructed.