Nevanlinna algebras

A. Haldimann¹ ; H. Jarchow¹

¹ Institut für Mathematik Universität Zürich Winterthurerstr. 190 CH-8057 Zürich, Switzerland

Studia Mathematica, Tome 147 (2001) no. 3, pp. 243-268

Résumé

The Nevanlinna algebras, ${\cal N}_\alpha ^p$, of this paper are the $L^p$ variants of classical weighted area Nevanlinna classes of analytic functions on ${\mathbb U}=\{ z\in {\mathbb C} :|z|1\} $. They are $F$-algebras, neither locally bounded nor locally convex, with a rich duality structure. For $s={(\alpha +2)/p}$, the algebra $F_s$ of analytic functions $f:{\mathbb U}\to {\mathbb C}$ such that $(1-|z|)^s|f(z)|\to 0$ as $|z|\to 1$ is the Fréchet envelope of ${\cal N}_\alpha ^p$. The corresponding algebra ${\cal N}_s^\infty $ of analytic $f:{\mathbb U}\to {\mathbb C}$ such that $\mathop {\rm sup}_{z\in {\mathbb U}}(1-|z|)^s|f(z)|\infty $ is a complete metric space but fails to be a topological vector space. $F_s$ is also the largest linear topological subspace of ${\cal N}_s^\infty $. $F_s$ is even a nuclear power series space. ${\cal N}_\alpha ^p$ and ${\cal N}_\beta ^q$ generate the same Fréchet envelope iff ${(\alpha +2)/p}={(\beta +2)/q}$; they can replace each other for quasi-Banach space-valued continuous multilinear mappings. Results for composition operators between ${\cal N}_\alpha ^p$'s can often be translated in a one-to-one fashion to corresponding ones on associated weighted Bergman spaces ${\cal A}_\alpha ^p$. This follows from the fact that the invertible elements in each ${\cal N}_\alpha ^p$ are precisely the exponentials of functions in ${\cal A}_\alpha ^p$. Moreover, each ${\cal N}_\alpha ^p$, ${(\alpha +2)/p}\le 1$, admits dense ideals. ${\cal A}_\alpha ^p$ embeds order boundedly into ${\cal A}_\beta ^q$ iff ${\cal A}_\beta ^q$ contains the Bloch type space ${\cal A}_{(\alpha +2)/p}^\infty $ iff ${(\alpha +2)/p}(\beta +1)/q$. In particular, $\bigcup _{p>0}{\cal A}_\alpha ^p$ and $\bigcap _{p>0}{\cal A}_\alpha ^p$ do not depend on the particular choice of $\alpha >-1$. The first space is a nuclear space, a copy of the dual of the space of rapidly decreasing sequences; the second has properties much stronger than being a Schwartz space but fails to be nuclear.

Détail

DOI : 10.4064/sm147-3-4

Mots-clés : nevanlinna algebras cal alpha paper variants classical weighted area nevanlinna classes analytic functions mathbb mathbb f algebras neither locally bounded nor locally convex rich duality structure alpha algebra analytic functions mathbb mathbb chet envelope cal alpha corresponding algebra cal infty analytic mathbb mathbb mathop sup mathbb infty complete metric space fails topological vector space largest linear topological subspace cal infty even nuclear power series space cal alpha cal beta generate chet envelope alpha beta replace each other quasi banach space valued continuous multilinear mappings results composition operators between cal alpha often translated one to one fashion corresponding associated weighted bergman spaces cal alpha follows the invertible elements each cal alpha precisely exponentials functions cal alpha moreover each cal alpha alpha admits dense ideals cal alpha embeds order boundedly cal beta cal beta contains bloch type space cal alpha infty alpha beta particular bigcup cal alpha bigcap cal alpha depend particular choice alpha first space nuclear space copy dual space rapidly decreasing sequences second has properties much stronger being schwartz space fails nuclear