Universal stability of Banach spaces for ε -isometries

• Autorzy: Lixin Cheng , Duanxu Dai , Yunbai Dong , Yu Zhou
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47A58: Operator approximation theory
• 46A20: Duality theory
• 46B20: Geometry and structure of normed linear spaces
• 46B04: Isometric theory of Banach spaces
• 26E25: Set-valued functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2014
• Tom: 221
• Numer: 2
• Strony: 141-149

Daty

wydano

2014

Twórcy

autor

Lixin Cheng

• School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

autor

Duanxu Dai

• School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

autor

Yunbai Dong

• School of Mathematics and Computer, Wuhan Textile University, Wuhan 430073, China

autor

Yu Zhou

• School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China

Identyfikatory

DOI

10.4064/sm221-2-3