Nevanlinna algebras

A. Haldimann ; H. Jarchow

Studia Mathematica, Tome 147 (2001), p. 243-268 / Harvested from The Polish Digital Mathematics Library

Résumé

The Nevanlinna algebras, $_{α}^{p}$, of this paper are the $L^{p}$ variants of classical weighted area Nevanlinna classes of analytic functions on = z ∈ ℂ: |z| < 1. They are F-algebras, neither locally bounded nor locally convex, with a rich duality structure. For s = (α+2)/p, the algebra $F_{s}$ of analytic functions f: → ℂ such that $(1-|z|)^{s}|f(z)| → 0$ as |z| → 1 is the Fréchet envelope of $_{α}^{p}$. The corresponding algebra $_{s}^{∞}$ of analytic f: → ℂ such that $sup_{z∈} (1-|z|)^{s} |f(z)| < ∞$ is a complete metric space but fails to be a topological vector space. $F_{s}$ is also the largest linear topological subspace of $_{s}^{∞}$. $F_{s}$ is even a nuclear power series space. $_{α}^{p}$ and $_{β}^{q}$ generate the same Fréchet envelope iff (α+2)/p = (β+2)/q; they can replace each other for quasi-Banach space-valued continuous multilinear mappings. Results for composition operators between $_{α}^{p}$’s can often be translated in a one-to-one fashion to corresponding ones on associated weighted Bergman spaces $_{α}^{p}$. This follows from the fact that the invertible elements in each $_{α}^{p}$ are precisely the exponentials of functions in $_{α}^{p}$. Moreover, each $_{α}^{p}$, (α+2)/p ≤ 1, admits dense ideals. $_{α}^{p}$ embeds order boundedly into $_{β}^{q}$ iff $_{β}^{q}$ contains the Bloch type space $_{(α+2)/p}^{∞} $ iff (α+2)/p < (β+1)/q. In particular, $⋃_{p>0}_{α}^{p}$ and $⋂_{p>0}_{α}^{p}$ do not depend on the particular choice of α > -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.

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Publié le : 2001-01-01

Zbl 0992.46017

EUDML-ID : urn:eudml:doc:285180