On Representations Of Coarse Isometries Of Banach Spaces

Let X,Y be Banach spaces,a mapping f:X?Y is said to be a coarse isometry provide that(?).In 1985,Lindenstrauss and Szankowski studied surjective coarse isometry and gave a nearly perfect answer.Because of some substantial difficullty,the nonsur-jective case has never been studied.This dissertation firstly considers isometric ap-proximation of coarse isometry in the sense of large distance and their weak stability.Thus,the problem that we consider is quite different from ever before.To explore the method of these issues,we take into account of some special Banach spaces,such as p uniformly convex spaces,Hilbert spaces and Lp spaces,we hope to inspect the concrete form and method of these problems on these spaces.Distinguished with Lin-denstrauss,Szankowski and Dolinar's method,which only dependent on the sequence limitation,we also use the filter limitation to deal the approximation and weak stabil-ity of a coarse isometry.The content is new.Main results of this dissertation are as follows:(?)When Y is a Hilbert space,we get a sufficient condition under which f can be approximated by a linear isometry,i.e.(?);(?)When Y is a p-uniformly convex space,if(?),then there exists a linear isometric approximation.Since Lp spaces are special q-uniformly convex spaces(q=max{2,p}),we obtain the results in Lp spaces;(?)When Y is a general Banach space,if(?),then f has an isometric approximation;(?)When f satisfies weak stability equation,then(?)defines an isometry from X to Y**;(?)We inspect the form of stability and weak stability of a coarse in the sense of filter limitation convergence.That is,f is u-weakly-stable if and only if(?)(?)defines an isometry.Combined with the special properties of Lp(1<p<?)spaces or Hilbert spaces,we get two equivalent full and sufficient conditions with which f is U weakly stable:(1)(?)is a linear isometry;(2)(?)exists.