On a property of Herglotz functions

Abstract

In this note I prove the following property of Herglotz functions, which to my knowledge is new: For a Herglotz function $h(z)$ and a real number $r\in \mathbb{R}$ define a Herglotz function $g_r(z)=(r-h(z){)}^{-1}.$ Let ${\mu }_r^{(s)}$ be the singular part of the measure ${\mu }_r$ which corresponds to $g_r(z)$ via the Herglotz representation theorem. Then the measure ${\int }_0^1{\mu }_r^{(s)}dr$ is absolutely continuous, its density is integer-valued a.e., and moreover the density takes values $0$ or $1$ a.e.

2 pages

References in corpus (1)

• Absolutely continuous and singular spectral shift functions