Nevanlinna algebras

A. Haldimann, and H. Jarchow. "Nevanlinna algebras." Studia Mathematica 147.3 (2001): 243-268. <http://eudml.org/doc/285180>.

@article{A2001,
abstract = {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.},
author = {A. Haldimann, H. Jarchow},
journal = {Studia Mathematica},
keywords = {weighted Nevanlinna classes; weighted Bergman spaces; Fréchet envelopes; nuclear power series spaces; topological algebras; composition operators; order boundedness; -algebra; absolutely summing operators},
language = {eng},
number = {3},
pages = {243-268},
title = {Nevanlinna algebras},
url = {http://eudml.org/doc/285180},
volume = {147},
year = {2001},
}

TY - JOUR
AU - A. Haldimann
AU - H. Jarchow
TI - Nevanlinna algebras
JO - Studia Mathematica
PY - 2001
VL - 147
IS - 3
SP - 243
EP - 268
AB - 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.
LA - eng
KW - weighted Nevanlinna classes; weighted Bergman spaces; Fréchet envelopes; nuclear power series spaces; topological algebras; composition operators; order boundedness; -algebra; absolutely summing operators
UR - http://eudml.org/doc/285180
ER -