Universal stability of Banach spaces for ε -isometries

Lixin Cheng, et al. "Universal stability of Banach spaces for ε -isometries." Studia Mathematica 221.2 (2014): 141-149. <http://eudml.org/doc/286160>.

@article{LixinCheng2014,
abstract = {Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T: L(f) ≡ \overline\{span\}f(X) → X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally right-stable spaces are just Hilbert spaces; every injective space is universally left-stable; a Banach space X isomorphic to a subspace of $ℓ_\{∞\}$ is universally left-stable if and only if it is isomorphic to $ℓ_\{∞\}$; and a separable space X has the property that (X,Y) is left-stable for every separable Y if and only if X is isomorphic to c₀.},
author = {Lixin Cheng, Duanxu Dai, Yunbai Dong, Yu Zhou},
journal = {Studia Mathematica},
keywords = {-isometry; stability; injective space; Banach space},
language = {eng},
number = {2},
pages = {141-149},
title = {Universal stability of Banach spaces for ε -isometries},
url = {http://eudml.org/doc/286160},
volume = {221},
year = {2014},
}

TY - JOUR
AU - Lixin Cheng
AU - Duanxu Dai
AU - Yunbai Dong
AU - Yu Zhou
TI - Universal stability of Banach spaces for ε -isometries
JO - Studia Mathematica
PY - 2014
VL - 221
IS - 2
SP - 141
EP - 149
AB - Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T: L(f) ≡ \overline{span}f(X) → X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally right-stable spaces are just Hilbert spaces; every injective space is universally left-stable; a Banach space X isomorphic to a subspace of $ℓ_{∞}$ is universally left-stable if and only if it is isomorphic to $ℓ_{∞}$; and a separable space X has the property that (X,Y) is left-stable for every separable Y if and only if X is isomorphic to c₀.
LA - eng
KW - -isometry; stability; injective space; Banach space
UR - http://eudml.org/doc/286160
ER -