On a universal inequality for approximate phase isometries

Let X and Y be two normed spaces. Let \({\cal U}\) be a non-principal ultrafilter on ℕ. Let g: X → Y be a standard ε-phase isometry for some ε ≥ 0, i.e., g(0) = 0, and for all u, v ϵ X,

$$|\,\,|\,||g(u) + g(v)|| \pm ||g(u) - g(v)||\,| - |\,||u + v|| \pm ||u - v||\,|\,\,|\, \le \varepsilon .$$

The mapping g is said to be a phase isometry provided that ε = 0. In this paper, we show the following universal inequality of g: for each \({u^ * } \in {w^ * } - \exp \,\,||{u^ * }||{B_{{X^ * }}}\), there exist a phase function \({\sigma _{{u^ * }}}:X \to \{ - 1,1\} \) and φ ϵ Y* with \(||\varphi || = ||{u^ * }|| \equiv \alpha \) satisfying that

$$|\left\langle {{u^ * },u} \right\rangle - {\sigma _{{u^ * }}}(u)\left\langle {\varphi ,g(u)} \right\rangle | \le {5 \over 2}\varepsilon \alpha ,\,\,\,{\rm{for}}\,{\rm{all}}\,u \in X.$$

In particular, let X be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all u* ∈ X*; (2) the constant \({5 \over 2}\) can be reduced to \({3 \over 2}\) provided that Y* is strictly convex; (3) the existence of such a g implies the existence of a phase isometry Θ: X → Y such that \(\Theta (u) = \mathop {\lim }\limits_{n,{\cal U}} {{g(nu)} \over n}\) provided that Y** has the w*-Kadec-Klee property (for example, Y is both reflexive and locally uniformly convex).