Research On The Isometric Operator And Phase-Isometric Operator On Non-Archimedean Normed Spaces

The non-Archimedean field,as a field,which is not isomorphic to either R or C.The study of non-Archimedean functional analysis is essential to other branches of mathematics,especially,in physics.It can be traced back to A.Monna,a Dutch mathematician in the 1940s.Since then,many scholars at home and abroad have conducted in-depth research on it.This article mainly studies the isometric and the phase-isometric problem of non-Archimedean normed space,and gets a positive answer to these questions.In the first chapter,we mainly describe the research background of this paper,which introduce the research progress of Mazur-Ulam Theorem and Tingley problem.And it gives the definition and research status of Mazur-Ulam property,and also introduce the research progress of Wigner’s Theorem.In the second chapter,we first introduce the non-Archimedean normed space.According to the property of non-Archimedean normed space,we describe isometric operators on Q3 in detail.On the basis of the existing conclusions,we further study the sufficient and necessary conditions of surjective isometry between spheres and get the following conclusions:Let X be a finite-dimensional non-Archimedean normed space.For any r∈‖X‖,every isometry fr:Sr(X)→Sr(Y)is surjective if and only if K is spherically complete and the residue class field k is finite.In the third chapter,we mainly study the phase-isometry of non-trivial non-Archimedean normed space,which can be phase equivalent to a linear isometric operator.That is,let X and Y be non-trivial non-Archimedean normed spaces,f:X→Y is a phase-isometry.Then there is a phase function ε:X→{-1,1} such that ε·f is an isometry.