Stability of isometries between the positive cones of ordered Banach spaces

Abstract

Given a real ordered Banach space E, denote its positive cone by $E_+$. A map** $T:E_+\rightarrow F_+$, where E and F are ordered Banach spaces, is said to be an $\varepsilon$-isometry if

$$\begin{aligned} \bigl |\Vert Tx-Ty\Vert - \Vert x-y\Vert \bigr | \le \varepsilon \text { for all }x,y \in E_+. \end{aligned}$$

T is $\delta$-surjective if for any $y\in F_+$, there exists $x\in E_+$ so that $\Vert y-Tx\Vert \le \delta$. In this paper, several classes of ordered Banach spaces E and F are identified so that the following is satisfied: There exists $K = K(E,F)<\infty$ such that for every $\delta$-surjective $\varepsilon$-isometry $T:E_+\rightarrow F_+$ with $T0=0$, there is a linear surjective isometry $\Phi :E\rightarrow F$ so that $\Phi (E_+) = F_+$ and that

$$\begin{aligned} \Vert Tx-\Phi x\Vert \le K\varepsilon .\end{aligned}$$

The classes for which this holds include all Banach lattices with strictly convex duals and the self-adjoint parts of the Schatten classes ($1<p<\infty$) and $C^*$-algebras, respectively. This is called Hyers–Ulam stability, which has been well investigated for $\varepsilon$-isometries $T:E\rightarrow F$ between Banach spaces. A stability result is also obtained for $\delta$-surjective $\varepsilon$-isometries $T:\ell ^1_+\rightarrow \ell ^1_+$. Examples are given to show that some of the constants K obtained in our results are best possible.