How to get universal inner functions
Many interesting results concerning universal inner functions have been proved recently by R.\,Aron, R.\,Mortini and the authors of this paper. Their purpose here is to present a result which puts the existence of universal inner functions in a more general framework. More precisely, a condition, called the mutual approximation property of the triple \((U,\mathcal{B},\mathcal{I})\) of a domain \(U\) in \(\mathbb{C}^d\), and subsets \(\mathcal{B}\) and \(\mathcal{I}\) of the closed unit ball of \(H^{\infty}(U)\), is introduced. The main theorem asserts that if \(\mathcal{I}\) is closed under multiplication and automorphisms, the constant function \(1\) belongs to both \(\mathcal{B}\) and \(\mathcal{I}\), and \((U,\mathcal{B},\mathcal{I})\) satisfies the mutual approximation property, then there is a closed infinite-dimensional vector space, generated by functions in \(\mathcal{I}\), isometric to \(\ell_1\), such that all its elements are universal. The assumptions are satisfied by the set \(\mathcal{I}\) of Blaschke products on the unit disc in \(\mathbb{C}\) and by the set of inner functions on the ball or, more generally, on a bounded symmetric domain. Therefore, the authors obtain as a consequence all known results and many new ones.
scientific article; zbMATH DE number 5189572
scientific article