| Let X and Y be real or complex normed spaces.A mapping f:X→Y is called aphase-isometry if it satisfifies the functional equation:{‖f(x)+f(y)‖,‖f(x)-f(y)‖}={‖x+y‖,‖x-y‖}((?)x,y ∈X)The first part of this paper mainly popularizes the famous Wigner’s theorem.By the famous Wigner’s theorem and Mazur-Ulam’s theorem,we show the phase-isometries in complex (?)~p(Γ)spaces(p≥1).The second part of this paper is inspired by the famous Tingley’s problem.The extension of isometries on the (?)~p(Γ)spaces’ unit spheres have been discussed and popularized.The first chapter mainly introduces the background and development status of Wigner’s theorem and Tingley’s problem.In addition,we give the definition of isometry,phase-isometry,phase equivalent,and point out the main problems in this paper.In the second chapter,we show the phase-isometries in complex (?)~p(Γ)spaces(P≥1).It is concluded that if the mapping f:(?)~p(Γ)→(?)~p(Δ)is a surjective phase-isometry,then there is a mapping ε:(?)~p(Γ)→(?)~p{-1,1} such that ε·f is a real linear isometry(also called phase equivalent to a real linear isometry).The phase-isometry in complex Hilbert spaces is also discussed separately.In the third chapter,we prove the extension of phase-isometries on the complex lP(Γ)spaces’ unit spheres.The following conclusion is:every surjective phase-isometries on the lP(Γ)spaces’ unit spheres can be extended to the whole spaces,and its positive homogeneous extension is also a phase-isometry which is phase equivalent a real linear isometry. |
