Three-space problems and bounded approximation properties

Wolfgang Lusky. "Three-space problems and bounded approximation properties." Studia Mathematica 159.3 (2003): 417-434. <http://eudml.org/doc/284452>.

@article{WolfgangLusky2003,
abstract = {Let $\{Rₙ\}_\{n=1\}^\{∞\}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_\{p\}$-space, then both X and A have bases. We apply these results to show that the spaces $C_\{Λ\} = \overline\{span\}\{z^\{k\} : k ∈ Λ\} ⊂ C()$ and $L_\{Λ\} = \overline\{span\}\{z^\{k\} : k ∈ Λ\} ⊂ L₁()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.},
author = {Wolfgang Lusky},
journal = {Studia Mathematica},
keywords = {three-space problems; basis projections},
language = {eng},
number = {3},
pages = {417-434},
title = {Three-space problems and bounded approximation properties},
url = {http://eudml.org/doc/284452},
volume = {159},
year = {2003},
}

TY - JOUR
AU - Wolfgang Lusky
TI - Three-space problems and bounded approximation properties
JO - Studia Mathematica
PY - 2003
VL - 159
IS - 3
SP - 417
EP - 434
AB - Let ${Rₙ}_{n=1}^{∞}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_{p}$-space, then both X and A have bases. We apply these results to show that the spaces $C_{Λ} = \overline{span}{z^{k} : k ∈ Λ} ⊂ C()$ and $L_{Λ} = \overline{span}{z^{k} : k ∈ Λ} ⊂ L₁()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.
LA - eng
KW - three-space problems; basis projections
UR - http://eudml.org/doc/284452
ER -