Hyers–Ulam stability of non-surjective isometries between subspaces of continuous functions
Volume 130 / 2023
Abstract
Let be locally compact Hausdorff spaces, and let C_0(X), C_0(Y) be the Banach spaces of all real continuous functions on X,Y, respectively, vanishing at infinity endowed with the usual sup-norm. Suppose that A, B are subspaces of C_0(X), C_0(Y), strongly separating points of X, Y, respectively; and denote by \partial A, \partial B the Shilov boundary of A,B, respectively. If T is a standard \varepsilon -isometry from A into B for some \varepsilon \gt 0, we prove that there is a non-empty subset Y_0 of \partial B, a surjective continuous map \tau : Y_0\rightarrow \partial _0 A (where \partial _0 A=\partial A \cap \mathrm {supp}(A)) and h\in C(Y_0) with |h(y)|=1 such that |h(y)f(\tau (y))-(Tf)(y)|\leq 3\varepsilon for all y\in Y_0 and f\in A. We also give an example to show the constant 3 in the above inequality is optimal.
