| Based on various characterizations of orthogonal correlation properties in inner product space,generalized orthogonality is established in normed linear space,among which Roberts orthogonality,Birkhoff orthogonality,isosceles orthogonal and so on have been studied most.Orthogonality-preserving operators are a class of operators that maintain some generalized orthogonality.Based on Mazur-Ulam theorem,scholars have studied distance preserving mappings in normed linear Spaces.Obviously,an isometric operator must be an orthogonality-preserving operator.But is the reverse ture? In 1992,Koldobsky demonstrated the conclusion that an orthogonality-preserving linear operator is a scalar multiple of an isometric on a real Banach space.Then some scholars have proved that the Birkhoff orthogonality-preserving linear operator is a scalar multiple of a linear isometric operator in general normed linear space.Based on the characterization of preserving orthogonal operators in inner product space,this paper focuses on the preservation of the properties of preserving orthogonal operators to point states.First proved the orthogonality-preserving operator in strictly convex space is a linear operator,and proves that T(x)is a not smooth point if and only if x is not smooth,which is Birkhoff orthogonality-preserving operator,on the basis of this again proves that if there is a Birkhoff orthogonality-preserving operator T between a two dimensional normed linear space X whose unit sphere is polygon and a normed linear space Y,then X and Y are isometric isomorphism.Finally,by using the concept of included Angle between vectors introduced by CHEN Zhi-zhi in Minkowski plane,a conclusion is drawn that linear Birkhoff orthogonality-preserving operator also preservs Angle. |
