Prime and semiprime inner functions (Q2869830)

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 and semiprime singular inner functions while the focus in the paper under review lies on several classes of Blaschke products. In order to state the main results, some more notations are necessary. A Blaschke product \(B\) is called thin if the sequence \((a_n)\) of zeros satisfies NEWLINE\[NEWLINE (1-|a_n|^2)|B'(a_n)| \to 1 \quad (n\to\infty)\,. NEWLINE\]NEWLINE If NEWLINE\[NEWLINE \sup_{\zeta \in \mathbb{T}}{\sum_n \frac{1-|a_n|}{|a_n-\zeta|}}<\infty\,, NEWLINE\]NEWLINE then \(B\) is called a uniform Frostman Blaschke product. Finally, for \(a \in \mathbb{D}\) let NEWLINE\[NEWLINE \varphi_a(z) = \frac{z-a}{1-\bar{a}z}\,. NEWLINE\]NEWLINENEWLINENEWLINEA first result on the denseness of Blaschke products states that the set of prime finite Blaschke products is dense in the set of all finite Blaschke products. Then, some results on semiprime and thin Blaschke products follow. If \(n\) is a positive integer, \(b(z)=z^n\) and \(B\) is a thin Blaschke product, then \(b \circ B\) is semiprime. The product of finitely many thin Blaschke products is semiprime. In particular, the product of a thin Blaschke product and a finite Blaschke product is semiprime. Section~5 of the paper deals with semiprimeness of inner functions with special Blaschke factors. If \(I\) is a nonconstant inner function whose Blaschke factor has distinct zeros, \(I(0)=0\), and \(I\) is continuous on a subarc \((\alpha,\beta) \subset \mathbb{T}\) of positive measure, then there is some positive integer \(n\) such that \(z^nI(z)\) is semiprime. Furthermore, there is a nonempty open set \(A \subset \mathbb{D}\) such that \(\varphi_a^2I\) is semiprime for all \(a \in A\). In particular, these results hold for uniform Frostman Blaschke products. If \(B\) is a thin Blaschke product, then there is a countable set \(S \subset \mathbb{D}\) such that \(\varphi_aB^n\) is prime for every \(a \in \mathbb{D} \setminus S\) and every positive integer \(n\). In Section~6 the authors prove results on prime approximations of finite products of thin Blaschke products. Finally, they provide some interesting examples: a surjective Blaschke product that is not semiprime, a Blaschke product with finite angular derivative everywhere that is not semiprime and a Blaschke product that is locally thin at a point \(m\) but not semiprime.