Universal stability of Banach spaces for $\varepsilon $-isometries

Lixin Cheng ¹   ; Duanxu Dai ¹   ; Yunbai Dong ²   ; Yu Zhou ³

¹ School of Mathematical Sciences Xiamen University Xiamen 361005, China
² School of Mathematics and Computer Wuhan Textile University Wuhan 430073, China
³ School of Fundamental Studies Shanghai University of Engineering Science Shanghai 201620, China

Studia Mathematica, Tome 221 (2014) no. 2, pp. 141-149

Résumé

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 $c_0$.

Détail

DOI : 10.4064/sm221-2-3

Keywords: real banach spaces varepsilon standard varepsilon isometry rightarrow said alpha gamma stable respect equiv mathop overline span nolimits rightarrow alpha gamma linear operator leq alpha tf uniformly bounded gamma varepsilon pair said stable every standard varepsilon isometry rightarrow alpha gamma stable alpha gamma space said universally right stable always stable every paper universally right stable spaces just hilbert spaces every injective space universally left stable banach space isomorphic subspace ell infty universally left stable only isomorphic ell infty separable space has property left stable every separable only isomorphic nbsp