This is an announcement for the paper "On $\eps$-isometry, isometry and linear isometry" by Lixin Cheng, Duanxu Dai, Yunbai Dong and Yu Zhou.
Abstract: Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $||f(x)-f(y)|-|x-y||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $\gamma>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$ such that $|Tf(x)-x|\leq\gamma\eps$ for all $x\in X$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show first that if such an $\eps$-isometry $f$ exists, then there is a linear isometry $U:X^{**}\rightarrow Y^{**}$. Then we prove that universally- right-stable spaces are just Hilbert spaces; every injective space is universally-left-stable; Finally, we verify that a Banach space $X$ which is linear isomorphic to a subspace of $\ell_\infty$ is universally-left-stable if and only if it is linearly isomorphic to $\ell_\infty$; and a separable space $X$ satisfying that $(X,Y)$ is stable for every separable $Y$ if and only if $X$ is linearly isomorphic to $c_0$.
Archive classification: math.FA
Mathematics Subject Classification: 46B04, 46B20, 47A58 (Primary) 26E25, 46A20, 46A24 (Secondary)
Remarks: 14 pages, submitted to Israel Journal of Mathematics
Submitted from: dduanxu@163.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1301.3374
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