Stability of isometries between the positive cones of ordered Banach spaces. (English) Zbl 07845309
Suppose \(E\) and \(F\) are ordered Banach spaces with positive cones \(E_{+}\) and \(F_{+}\). Let \(T:E_{+} \rightarrow F_{+}\) be an \(\varepsilon\)-isometry, i.e.,
\[
|\|T(x)-T(y)\|-\|x-y\||\leq\varepsilon \ \ \ \forall x,y\in E_{+}.
\]
Assume further that \(T\) is \(\delta\)-surjective, i.e., for all \(y\in F_{+}\) there is some \(x\in E_{+}\) such that \(\|T(x)-y\|\leq\delta\), and that \(T(0)=0\).
The authors consider the following Hyers-Ulam stability problem: Is there a linear surjective isometry \(\Phi:E \rightarrow F\) such that \(\Phi(E_{+})=F_{+}\) and \[ \|T(x)-\Phi(x)\|\leq K\varepsilon \ \ \ \forall x\in E_{+} \] with a constant \(K\) depending only on \(E\) and \(F\)?
The answer is shown to be affirmative for a wide class of ordered Banach spaces, including for instance all Banach lattices with strictly convex dual and the self-adjoint parts of \(C^*\)-algebras.
The case of \(\delta\)-surjective \(\varepsilon\)-isometries \(T:\ell^1_{+} \rightarrow \ell^1_{+}\) is also treated by a different method.
The authors consider the following Hyers-Ulam stability problem: Is there a linear surjective isometry \(\Phi:E \rightarrow F\) such that \(\Phi(E_{+})=F_{+}\) and \[ \|T(x)-\Phi(x)\|\leq K\varepsilon \ \ \ \forall x\in E_{+} \] with a constant \(K\) depending only on \(E\) and \(F\)?
The answer is shown to be affirmative for a wide class of ordered Banach spaces, including for instance all Banach lattices with strictly convex dual and the self-adjoint parts of \(C^*\)-algebras.
The case of \(\delta\)-surjective \(\varepsilon\)-isometries \(T:\ell^1_{+} \rightarrow \ell^1_{+}\) is also treated by a different method.
Reviewer: Jan-David Hardtke (Berlin)
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