Universal stability of Banach spaces for $\varepsilon $-isometries
Volume 221 / 2014
Abstract
Let $X$, $Y$ be real Banach spaces and $\varepsilon >0$. A standard $\varepsilon $-isometry $f:X\rightarrow Y$ is said to be $(\alpha ,\gamma )$-stable (with respect to $T:L(f)\equiv \mathop {\overline {\rm span}}\nolimits f(X)\rightarrow X$ for some $\alpha , \gamma >0$) if $T$ is a linear operator with $\| T\| \leq \alpha $ such that $Tf-{\rm Id}$ is uniformly bounded by $\gamma \varepsilon $ on $X$. The pair $(X,Y)$ is said to be stable if every standard $\varepsilon $-isometry $f:X\rightarrow Y$ is $(\alpha ,\gamma )$-stable for some $\alpha ,\gamma >0$. 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 $\ell _\infty $ is universally left-stable if and only if it is isomorphic to $\ell _\infty $; 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 $c_0$.
