A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices
type: Item
instance of: scholarly article
title: A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices
zbMATH Open document ID: 1064.30030
author: B. Simon
published in: Journal of Functional Analysis
publication date: 2004-10-01T00:00:00Z
review text: Let \(f\) be a function meromorphic in the unit disc \(D\) and, for \(\zeta \in D\) let \[ b(z, \zeta) = \frac {| \zeta| } {\zeta} \cdot \frac {\zeta - z} {1 - \bar{\zeta} z} \quad \zeta \neq 0, \] and \(b(z, 0) = z\). The author proves that if \(f\) is real valued on the interval \((-1, 1)\) and \(\Im f(z) > 0\) for \(\Im z > 0\), then, \[ f(e^{i\theta}) = \lim_{r \to 1} f(r e^{i\theta}) \;\text{ exists for almost every } \theta \in [0, 2\pi). \] If, in addition, \[ \int_{0}^{2\pi} \;| \log{| f(e^{i\theta}| }| ^{p} \frac {d\theta} {2\pi} < \infty \] for all \(p < \infty\), and if \(p_{1}^{+} < p_{2}^{+} < \dots\) and \(z_{1}^{+} < z_{2}^{+} \dots\) are the poles and zeros, respectively, of \(f\) in the interval \([0, 1)\) and if \(p_{1}^{-} > p_{2}^{-} > \dots\) and \(z_{1}^{-} > z_{2}^{-} > \dots\) are the poles and zeros, respectively, of \(f\) in the interval \((-1, 0)\), then \[ B(z) = \lim_{n \to \infty} \prod_{j=1}^{n} b(z, z_{j}^{+}) (b(z, p_{j}^{+}))^{-1} b(z, z_{j}^{-}) (b(z, p_{j}^{-}))^{-1} \] converges uniformly on compact subsets of \(D\) that do not contain poles of \(f\), and \[ f(z) = \pm B(z) \exp{\bigg(\int_{0}^{2\pi} \frac {e^{i\theta} + z} {e^{i\theta} - z} \log{| f(e^{i\theta})| } \frac {d\theta} {2\pi} \bigg)} \;, \] where the \(\pm\) sign in front is \(+\) if \(f(0) = 0\) and the sign of \(f(0)\) if \(f(0) \neq 0\). Note that the zeros and poles are not required to be Blaschke sequences, but the hypothesis guarantees that each zero and pole is simple and that the zeros and poles alternate on the real axis. This last is what guarantees the convergence of \(B(z)\). This result leads to a significant simplification of a result due to \textit{R. Killip} and the author [Ann. Math. (2) 158, 253--321 (2003; Zbl 1050.47025)] giving a formula satisfied by the spectral measures of Jacobi matrices \(J\) with \(J - J_{0}\) Hilbert-Schmidt.
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 47B36
zbMATH DE Number: 2105317
zbMATH Keywords: factorization of meromorphic functions
zbMATH Keywords: Jacobi matrices
reviewed by: Peter Lappan
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.11.006
OpenAlex ID: W1981352821
cites work: Jost functions and Jost solutions for Jacobi matrices, II. Decay and analyticity
cites work: Q5832957
cites work: Q4241837
cites work: Q3752135
cites work: Sum rules for Jacobi matrices and their applications to spectral theory
cites work: Q4340161
cites work: Aizenman's theorem for orthogonal polynomials on the unit circle
cites work: Sum rules and the Szegő condition for orthogonal polynomials on the real line
DOI: 10.1016/J.JFA.2003.11.006
description: scientific article; zbMATH DE number 2105317