Stability of symmetric $$\varepsilon $$ ε -isometries on wedges

Let X, Y be Banach spaces, W be a closed wedge of X, and $$f:W\cup -W\rightarrow Y$$ f : W ∪ - W → Y be a symmetric $$\varepsilon $$ ε -isometry. We firstly establish a weak stability formula about the symmetric isometry f. Making use of it, we prove a series of new stability theorems for the symmetric isometries defined on the positive cones W of C(K)-spaces. For example, if $$f(W\cup -W)$$ f ( W ∪ - W ) contains a reproducing wedge of Y, then there exists a linear surjective isometry $$U:C(K)\rightarrow Y$$ U : C ( K ) → Y such that $$f-U$$ f - U is uniformly bounded by $$\frac{3}{2}\varepsilon $$ 3 2 ε on $$W\cup -W$$ W ∪ - W ; and if $$\overline{\mathrm{co}}f(W\cup -W)$$ co ¯ f ( W ∪ - W ) contains a reproducing wedge P of Y, then there exists a bounded linear operator $$T:Y\rightarrow C(K)$$ T : Y → C ( K ) with $$\Vert T\Vert =1$$ ‖ T ‖ = 1 such that $$\begin{aligned} \Vert Tf(x)-x\Vert \le \frac{3}{2}\varepsilon ,~~{\mathrm{for\;\; all\;}}~ x\in W\cup -W. \end{aligned}$$ ‖ T f ( x ) - x ‖ ≤ 3 2 ε , for all x ∈ W ∪ - W .