Phase-isometries On Some Special Vector-valued Function Spaces

The famous Wigner's theorem plays an exceedingly significant role in quantum mechanics and in representation theory in physics,it expresses the symmetry transformation in quantum mechanics by mathematical language.Based on Wigner theorem,let X and Y be real normed spaces.We say that the mapping f:X?Y is called a phase-ismoetry if it satisfies the functional equationInspired by open problems from Maska and Páles and Mazur-Ulam theorem,we can consider the following Wigner-type problem:Let X and Y be real normed spaces.Suppose that f:X?Y is a surjective phase-isometry.Is it true that f is a phase equivalent to a real linear isometry,namely,is there a phase functione:X?{-1,1}such thate×f is a real linear isometry?In this paper,we mainly discuss phase-isometry problems on two special types of vector-function spaces.Namely,we give positive answers to study phase-isometry problem on real (?)space and real (?)-type space.In first chapter,we mainly introduce the background and development of Mazur-Ulam theorem and Wigner theorem,give the definition of phase-isometry and phase eqivalent,and raise the problems which we need to study.In second chapter,we study phase-isometry problem on real (?)spaces and give the following conclusion:Let(?)be nonempty index sets and H,K be real inner spaces,p31.Then surjective phase-isometry mapping is phase equivalent to a real linear isometry from (?).In third chapter,we study phase-isometry problem on real (?)-type spaces and give the following conclusion:Let(?)be nonempty index sets and E,F be real strictly convex normed spaces.Then surjective phase-isometry mapping is phase equivalent to a real linear isometry from (?).Moreover,if H is a real inner space,the real (?)-type space has the Wigner property.Namely,for any normed space Y,surjective phase-isometry is phase equivalent to a real linear isometry from (?)to Y.