改进的Tsirelson空间TM上的Wigner定理

Wigner's Theorem on the Modified Tsirelson Space TM

设X和Y是赋范空间,如果映射f:X→Y满足{||f(x)+f(y)||,||f(x)-f(y)||}={||x+y||,||x-y||}(x,y∈X),则称f是一个相位等距算子.设g,f:X→Y是映射,若存在相位函数ε:X→{-1,1},使得ε·f=g,则称g和f是相位等价的.本文将证明改进的Tsirelson空间TM上的任意满相位等距算子均相位等价于一个线性等距算子.该结论同时也给出了改进的Tsirelson空间TM上的Wigner型定理.

Let X and Y be normed space.We say that a map f:X→Y is a phaseisometry if it satisfies{||f(x)+f(y)||,||f(x)-f(y)||}={||x+y||,||x-y||}(x,y∈X).Suppose that g,f:X→Y are maps.If there is a phase functionε:X→{-1,1}such thatε.f=g,then we say that f is phase-equivalent to g.We shall prove thatevery phase-isometry between two modified Tsirelson spaces TM is phase-equivalent toa linear isometry.This can be considered as a new version of the famous Wigner'stheorem for the modified Tsirelson space TM.