Research On Extension Of Phase-Isometries Between The Unit Spheres Of L^∞（Γ,H） Space

Let X and Y be real or complex normed spaces,if the mapping f:X→Y,for any x,y ∈X{‖ f（x）+f（y）‖,‖ f（x）-f（y）‖}={‖x+y‖,‖ x-y ‖}Then f is a phase-isometry mapping.In this paper,the famous Wigner’s theorem is generalized,combined with the Wigner’s theorem and the Tingley problem,to prove the continuation problem of its phase equidistant on the unit sphere of real and complex lp（Γ,H）space respectively.In the first chapter,the development history and research status of Wigner’s theorem,Mazur-Ulam’s theorem and Tingley problem are mainly introduced,and the relevant conclusions of isometric and phase isometric are given.In the second chapter,we mainly study the extension of phase-isometries on the real lp（Γ,H）space unit sphere,and finally come to the conclusion:the surjective phase-isometry on the real lp（Γ,H）space unit sphere can be extended to the whole space,and its natural extension is The topological phase is equivalent to a real linear equidistant.In the third chapter,we mainly study the extension of phase-isometries on the complex l∞（Γ,H）space unit sphere.The research method is based on the Wigner-type theorem in real l∞（Γ,H）space and the extension of phase-isometries on the unit sphere of complex l∞（Γ）space.Different situations in complex space are studied,and the following conclusions are obtained:the mapping on the complex l∞（Γ,H）space unit sphere can be extended to a surjective phase-isometry in the whole space,and this mapping is phase equivalent to a real linear equidistant mapping.