Hyers–Ulam stability of non-surjective isometries between subspaces of continuous functions

Let $X, Y$ 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.