Krein-type theorems and ordered structure for Cauchy-de Branges spaces (Q2416592)

An entire function \(f\) is said to be of exponential type \(c>0\) if for every \(\epsilon>0\) there exists a constant \(A_\epsilon\) such that \[|f(z)|\le A_\epsilon e^{(c+\epsilon)|z|}\] for \(|z|\to \infty\). In a similar fashion, an entire function \(F\) is said to be of \textit{Cartwright class} if it is of finite exponential type and satisfies \[\int_{\mathbb {R}}\frac{\log ^+ |F(x)|}{1+x^2}dx <\infty.\] A function \(f\) analytic in the upper half-plane \(\mathbb{C}^+\) is called of bounded type if there exist two bounded analytic functions \(g\) and \(h\) such that \(f=\frac{g}{h}\). The number \[mt(f)=\limsup_{y\to \infty}\frac{|\log f(iy)|}{y}\] is called the mean type of \(f\).\par Let us begin by recalling a theorem of M.G. Krein: Let \(F\) be an entire function that is of bounded type both in the upper half-plane \(\mathbb{C}^+\) and in the lower half-plane \(\mathbb{C}^-\). Then \(F\) is a function of Cartwright class. In particular, \(F\) is of finite exponential type, and its type equals the maximum of mean types of \(F\) in the upper and lower half-planes. A typical situation for the application of Krein's theorem is the case when \(F\) can be represented as the ratio of two discrete Cauchy transforms \[F=\frac{\mathcal{C}_{a,k}(z)}{\mathcal{C}_{b,m}(z)},\] where \(\mathcal{C}_{a,k}(z)\) is defined in the following way: Let \((t_n)\) be a sequence of non-zero real numbers, and \(a=(a_n)\) be an absolutely convergent sequence. The \textit{discrete Cauchy transform} is defined by \[\mathcal{C}_a(z)=\sum_n\frac{a_n}{z-t_n},\] and if for some \(k\in\mathbb{N}\) we have \[\sum_n\frac{|a_n|}{|t_n|^{k+1}}<\infty,\] then the \textit{regularized Cauchy transform} is defined by \[\mathcal{C}_{a,k}(z)=z^k \sum_n\frac{a_n}{t_n^k(z-t_n)}.\] By Krein's theorem, every \(F=\frac{\mathcal{C}_{a,k}(z)}{\mathcal{C}_{b,m}(z)}\) is of finite exponential type.\par Another result of Krein states that if \(F\) is an entire function, which is real on \(\mathbb{R}\), with simple zeros \(t_n\neq 0\) on the real line, and such that for some non-negative integer \(k\) we have \[\sum_n\frac{1}{|t_n|^{k+1}|F^\prime (t_n)|}<\infty,\] and \[\frac{1}{F(z)}=R(z)+\sum_n\frac{1}{F^\prime (t_n)}\left(\frac{1}{z-t_n}+\frac{1}{t_n}+\cdots+\frac{z^{k-1}}{t_n^k}\right ),\] where \(R\) is some polynomial, then \(F\) is of Cartwright class.\par The main objective of the paper under review is to extend the above results of Krein to the class of Cauchy transforms of measures that are supported by some discrete set \(\{t_n\}\) in the complex plane where \(t_n\) are no longer real numbers. It is of course assumed that the points \(t_n\) are pairwise distinct and \(|t_n|\to \infty\) as \(n\to \infty\).\par More precisely, the authors discuss the following three cases for the sequence \(T=(t_n)\). (i) \(T\) is the zero set of some entire function of zero exponential type; (ii) \(T\) lies in some strip and has finite convergence exponent; (iii) \(T\) lies in some angle of size \(\gamma \pi,\, 0<\gamma<1\), and the convergence exponent of \(T\) is strictly less than \(\gamma^{-1}\). As an application, the authors obtain a new version of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of analytic functions.