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

Let $X$, $Y$ be two real Banach spaces and $\epsilon >0$. A standard $\epsilon$-isometry $f:X\to Y$ is said to be $(\alpha ,\gamma )$-stable (with respect to $T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$forsome$\alpha, \gamma>0$)if$T$ is a linear operator with $\mathrm{\parallel }T\mathrm{\parallel }\le \alpha$ so that $Tf-Id$ is uniformly bounded by $\gamma \epsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every standard $\epsilon$-isometry $f:X\to Y$ is $(\alpha ,\gamma )$-stable for some $\alpha ,\gamma >0$. $X(Y)$ 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 ${\mathrm{\ell }}_{\mathrm{\infty }}$ is universally left-stable if and only if it is isomorphic to ${\mathrm{\ell }}_{\mathrm{\infty }}$; 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 $c_0$.