A note on stability of non-surjective $\varepsilon$-isometries between the positive cones of $L^p$-spaces

Abstract

Let $(\Omega _i, \Sigma _i, \mu _i)$, $i=1, 2,$ be two measure spaces, $1<p<\infty$, $X_i=L^p(\Omega _i, \Sigma _i, \mu _i)$, and let $X_i^+=\{f \in X_i: f\ge 0\; a.e.\}$ be the positive cone of $X_i$. In this paper, we first show that for every standard $\varepsilon$-isometry $F: X_1^+\rightarrow X_2^+$, there exists a bounded linear surjective operator $T:X_2\rightarrow X_1$ with $\Vert T\Vert =1$ such that

$$\begin{aligned} \Vert TF(u)-u\Vert \le 4\varepsilon , \mathrm{\;for\;all\;} u\in X_1^+. \end{aligned}$$

As its application, we prove that if $\liminf \limits _{t\rightarrow +\infty }\mathrm{dist}(ty,F(X_1^+))/t<1/2$ for all $y\in S_{X_2^+}$, then there exists a unique additive surjective isometry $V: X_1^+\rightarrow X_2^+$ so that

$$\begin{aligned} \Vert F(u)-V(u)\Vert \le 4\varepsilon ,\mathrm{\;for\;all\;} u\in X_1^+. \end{aligned}$$

We also unify some results concerning the Hyers-Ulam stability of almost surjective $\varepsilon$-isometries between the positive cones of two $L^p$-spaces.