On Stability and Weak-Star Stability of $$\varepsilon $$-Isometries

Abstract

Let X, Y be two Banach spaces, $$f:X\rightarrow Y$$ be an $$\varepsilon $$ -isometry with $$f(0)=0$$ for some $$\varepsilon \ge 0$$ , and let $$Y_f\equiv \overline{\mathrm{span}}f(X)$$ . In this paper, we first introduce a notion of $$w^*$$ -stability of an $$\varepsilon $$ -isometry f. Then we show that stability of f implies its $$w^*$$ -stability; the two notions of stability and $$w^*$$ -stability coincide whenever X is a dual space and they are not equivalent in general. Making use of a recent sharp weak stability estimate of f, we then improve some known results.