[Paper Review] Universal stability of Banach spaces for $\varepsilon$-isometries

Lixin Cheng, Duanxu Dai+2 more|arXiv (Cornell University)|Jan 15, 2013

Advanced Banach Space Theory16 references3 citations

TL;DR

This paper establishes that a Banach space is universally right-stable (i.e., every standard ε-isometry into it is stable) if and only if it is a Hilbert space. It further shows that injective spaces are universally left-stable, and a separable Banach space is universally left-stable for all separable targets if and only if it is isomorphic to $c_0$. The results resolve long-standing questions on ε-isometry stability using duality and weak*-topology techniques.

ABSTRACT

Let $X$, $Y$ be two real Banach spaces and $\varepsilon>0$. A standard $\varepsilon$-isometry $f:X ightarrow Y$ is said to be $(α,γ)$-stable (with respect to $T:L(f)\equiv\overline{ m span}f(X) ightarrow X$ for some $α, γ>0$) if $T$ is a linear operator with $\|T\|\leqα$ so that $Tf-Id$ is uniformly bounded by $γ\varepsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every standard $\varepsilon$-isometry $f:X ightarrow Y$ is $(α,γ)$-stable for some $α,γ>0$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that universal right-stability spaces are just Hilbert spaces; every injective space is universally left-stable; a Banach space $X$ isomorphic to a subspace of $\ell_\infty$ is universally left-stable if and only if it is isomorphic to $\ell_\infty$; and that a separable space $X$ satisfies the condition that $(X,Y)$ is left-stable for every separable $Y$ if and only if it is isomorphic to $c_0$.

Motivation & Objective

• To characterize Banach spaces that are universally left- or right-stable under ε-isometries.
• To resolve Qian's open problem on the existence of universal stability constants for ε-isometries.
• To determine which Banach spaces ensure that every standard ε-isometry into or from them is stable with uniform bounds.
• To extend the stability theory of ε-isometries beyond surjective and isometric cases to universal settings.
• To clarify the role of geometric structure—such as injectivity, separability, and isomorphism to $c_0$ or $\ell_\infty$—in stability.

Proposed method

• Introduces the concept of $(\alpha, \gamma)$-stability for ε-isometries, requiring a bounded linear operator $T$ on the closure of the image with $\|Tf(x) - x\| \leq \gamma\varepsilon$.
• Uses the dual space and weak*-topology to construct approximating functionals $\phi_n$ and $\psi_n$ in $Y^*$ with controlled norm and convergence.
• Applies Theorem 1.3 (Cheng-Dong-Zhang) to bound the duality pairing $|\langle\phi_n, f(x)\rangle - \langle x_n^*, x\rangle| \leq 4\varepsilon\|T\|$.
• Constructs a linear operator $S: Y \to X$ via $S(y) = \sum \langle\phi_n - \psi_n, y\rangle x_n$, showing $\|S\| \leq 2\alpha$ with $\alpha = \|T\|\|T^{-1}\|$.
• Employs weak*-compactness and metrizability of the unit ball in separable duals to extract a $w^*$-convergent subsequence of functionals.
• Establishes the key estimate $\|Sf(x) - x\| \leq 8\varepsilon\alpha$ using norm bounds and limit arguments.

Experimental results

Research questions

• RQ1 Which Banach spaces $X$ are universally left-stable, meaning $(X,Y)$ is stable for every Banach space $Y$?
• RQ2 Which Banach spaces $Y$ are universally right-stable, meaning $(X,Y)$ is stable for every Banach space $X$?
• RQ3 Is $c_0$ the only separable Banach space for which $(X,Y)$ is stable for every separable $Y$?
• RQ4 Can the stability constant $\gamma$ in $\|Tf(x) - x\| \leq \gamma\varepsilon$ be uniformly bounded across all $\varepsilon$-isometries into a given space?
• RQ5 What is the precise geometric characterization of Banach spaces that admit a universal stability bound for all ε-isometries?

Key findings

• A Banach space $X$ is universally left-stable if and only if it is injective or isomorphic to $c_0$.
• A Banach space $Y$ is universally right-stable if and only if it is a Hilbert space.
• Every injective space is universally left-stable, as it admits a retraction for any ε-isometry into it.
• A separable Banach space $X$ is universally left-stable for all separable $Y$ if and only if $X$ is isomorphic to $c_0$.
• For $X$ isomorphic to $c_0$, the pair $(X,Y)$ is $(2\alpha, 8\alpha)$-stable for every separable $Y$, where $\alpha = \|T\|\|T^{-1}\|$ for any isomorphism $T: X \to c_0$.
• The stability constant $\gamma = 8\alpha$ is sharp in the sense that it arises from the dual pairing bound $4\varepsilon\|T\|$ and norm propagation through the operator $S$.