ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

Let X and Y be two normed spaces.Let U be a non-principal ultrafilter on N.Let g:X→Y be a standard ε-phase isometry for someε≥ 0,i.e.,g(0)=0,and for all u.v ∈ X,||‖g(u)+g(v)‖±‖g(u)-g(v)‖|-|‖u+v‖±‖u-v‖| |≤ε.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^(*) ∈ w^(*)-exp ‖u^(*)‖B_(x^(*)),there exist a phase function σ_(u^(*)):X→{-1,1} and φ ∈ Y^(*) with ‖φ‖=‖u^(*)‖≡α satisfying that|(u^(*),u)-σ_(u^(*))(u)<φ,g(u)>)|≤5/2εα,for all u ∈ 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 5/2 can be reduced to 3/2 provided that Y~*is strictly convex;(3) the existence of such a g implies the existence of a phase isometryΘ:X→Y such that■ provided that Y^(**) has the w^(*)-Kadec-Klee property(for example,Y is both reflexive and locally uniformly convex).