A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices
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.
scientific article; zbMATH DE number 2105317
scientific article