Prime and semiprime inner functions
type: Item
instance of: scholarly article
title: Prime and semiprime inner functions
zbMATH Open document ID: 1295.30129
author: Pamela Gorkin
author: Isabelle Chalendar
author: Jonathan R. Partington
publication date: 2014-01-07T00:00:00Z
Mathematics Subject Classification ID: 30J05
Mathematics Subject Classification ID: 30J10
Mathematics Subject Classification ID: 46J20
zbMATH DE Number: 6243086
zbMATH Keywords: inner function
zbMATH Keywords: Blaschke product
zbMATH Keywords: primeness
zbMATH Keywords: semiprimeness
Wikidata QID: Q56319176
MaRDI profile type: Publication
full work available at URL: jdt038
OpenAlex ID: W2107370227
DOI: 10.1112/JLMS/JDT038
published in: Journal of the London Mathematical Society. Second Series
review text: Let \(\mathbb{D}\) denote the open unit disk in the complex plane \(\mathbb{C}\). A holomorphic function \(u : \mathbb{D} \to \overline{\mathbb{D}}\) is called an inner function if it has radial limits of modulus \(1\) at almost every point of the unit circle \(\mathbb{T} = \{\, z \in \mathbb{C} : |z|=1 \,\}\). Examples of inner functions are Blaschke products (especially Möbius transformations) and singular inner functions. Moreover, every inner function \(u\) can be factorized in the form \(u=BS\) with a Blaschke product \(B\) and a singular inner function \(S\). This factorization is unique up to a constant of modulus \(1\). If \((a_n)\) denotes the sequence of zeros of \(u\) (counting multiplicities), then the Blaschke condition NEWLINE\[NEWLINE \sum_n (1-|a_n|)<\infty NEWLINE\]NEWLINE is satisfied, and the Blaschke product associated with \((a_n)\) is defined by NEWLINE\[NEWLINE B(z) = \prod_n \frac{|a_n|}{a_n}\frac{a_n-z}{1-\bar{a}_nz}\,, \quad z \in \mathbb{D}, NEWLINE\]NEWLINE NEWLINENEWLINEwith the convection that \(|a_n|/a_n=1\) if \(a_n=0\). Such a Blaschke product is called finite if it has only finitely many zeros. The singular inner factor \(S=S_\mu\) associated with a finite positive Borel measure on \(\mathbb{T}\) singular with respect to Lebesgue measure, is defined by NEWLINE\[NEWLINE S_\mu(z) = \exp{\left(-\int_0^{2\pi} \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\mu(\theta)\right)}\,, \quad z \in \mathbb{D}\,. NEWLINE\]NEWLINE The composition of two inner functions is still an inner function. An inner function \(I\) is called prime (with respect to composition) if whenever \(I = u \circ v\) with inner functions \(u\) and \(v\), then \(u\) or \(v\) is a Möbius transformation. More generally, \(I\) is called semiprime if whenever \(I = u \circ v\) with inner functions \(u\) and \(v\), then \(u\) or \(v\) is a finite Blaschke product.NEWLINENEWLINE\textit{P.~Gorkin} et al. [Result. Math. 25, No. 3--4, 252--269 (1994 Zbl 0799.30023)] mainly studied prime a…
description: scientific article; zbMATH DE number 6243086