Hyers--Ulam stability of weighted composition operators on \(L^p\)-spaces (Q2460756)

Recall that a mapping \(T\) from a Banach space \(B\) into itself is said to display Hyers--Ulam stability if there exists \(K>0\) such that, for any \(g\in T(B)\), \(\varepsilon>0\) and \(f\in B\) such that \(\| Tf-g\| <\varepsilon\), we can find \(f_0\in B\) with \(Tf_0=g\) and \(\| f-f_0\| \leq K\varepsilon\). The author analyzes when the weighted composition operators \(uC_\phi\) acting on \(L^p(\Sigma)\) displays Hyers--Ulam stability. Let \((X,\Sigma,\mu)\) be a complete \(\sigma\)-finite measure space and \(\phi:X\to X\) a non-singular measurable function (that is, \(\mu(A)=0\) implies \(\mu(\phi^{-1}(A))=0\)) such that \(\Sigma\) is \(\phi\)-invariant (that is, \(\phi(A)\in \Sigma\) for any \(A\in \Sigma\)) and \(\mu\) is normal (that is, \(\mu(\phi(A))=0\) if \(\mu(A)=0\)). The main result establishes that the operator \(uC_\phi: L^p(\Sigma)\to L^p(\Sigma)\), \(1\leq p<\infty\), displays Hyers--Ulam stability if and only if there exists \(r>0\) such that \(J(x):=(h(x) E(| u| ^p)\circ\phi^{-1}(x))^{1/p}\geq r\) \(\mu\)-a.e.\ in the support of \(J\), where \(h=\frac{d\mu\circ \phi^{-1}}{d\mu}\) or, equivalently, \(\phi(\{u\neq 0\})\subset\{J\geq r\}\).