The Extension Of Isometries And Nonexpansive Maps On The Unit Spheres

The well known Mazur-Ulam theorem states that every surjective isometry be-tween two real normed spaces which takes 0 to 0 is necessarily linear. Tingley first interested isometries defined only on the unit sphere of the normed space. He raised a general:Problem 1. Let E and F be normed spaces with unit spheres S(E) and S(F). Suppose that V:S(E)→S(F) is a surjective isometry. Is there a linear isometric mapping V:E→F such that V|s(E) = V?It is called Tingley’s problem or the isometric extension problem.On the other hand, an observation that every surjective nonepansive map from from a compact metric space onto itself must be an isometry brings the study of another problem which has a close relation with Tingley’s problem and can be described as follows:Problem 2. Let E be an infinite-dimensional Banach space or only an F-space, and let T:S(E)→S(E) be a surjective nonexpansive map. Is T necessarily an isometry?The main theme of this dissertation is the study of Problem 1 (main) and Prob-lem 2 which is naturally connected to the study of the geometry and structure of Banach spaces or generally F-spaces.Firstly we give a detailed introduction of the background and the existent results of Problem 1 and Problem 2 in Chapter 1.In Chapter 2, we prove the representation theorems of surjective isometries be-tween the unit spheres of the generalized James space Jp and The Tsirelson space T. By the representation theorems, we give an affirmative answer to Tingley’s problem for the spaces Jp and T.In Chapter 3, we consider Tingley’s problem in a more general condition. To be precise, we consider the surjective isometries between the unit spheres of the Lp-spaces (1≤p≤∞) and a Banach space E. We show that such isometries can be extended to a linear isometry on the whole space. Additionally, the isometric extension problem of the isometries between the unit spheres of Lp(μ) and a linear subspace of Lp(μ) in the case 0< p< 1 is studied, and we also obtain a positive answer to Tingley’s problem in such spaces.In Chapter 4, we focus on the study of Problem 2. We prove that the only non-expansive mappings from the unit sphere of L∞(г) type spaces (including c00, c,l∞) (resp. from lp(г)(0< p≤1)) onto the unit sphere of L∞(Δ) (resp. onto lp(Δ)(0< p≤1)) are those arising from a bijection betweenΔandгand a sign pattern. This result yields a fact that such maps are isometries and an affirmative answer to Tingley’s problem for such spaces.In Chapter 5, we consider maps which preserves equality of distance. Vogt proved that such map of normed spaces which takes 0 to 0 is necessarily linear. In this chapter, we extend this result to the class of F*-spaces including all p-normed spaces(0<p≤1).