On Maps Preserving Generalized Orthogonality Types

In inner product spaces, maps preserving orthogonality must be a scalar multiple of a linear isometry. One natural question is whether or not this result still holds in general normed linear spaces. Research on maps preserving some orthogonality helps us to understand properties of this orthogonality and its influence on the properties of the underlying space.Firstly, we survey the introductions of some types of orthogonality, existing results on properties of orthogonality types, relation between two different orthogonality types, and maps preserving generalized orthogonalities, and characterizations of inner product spaces related to generalized orthogonalities.Secondly, we collect the definitions of bilinear form, antinorm, semi-inner product, and the lower and upper semi-inner products, present the relation between norm orthogonality and Birkhoff orthogonality, and between Lumer orthogonality and Birkhoff orthogonality.Finally, as the main results in this paper, we present a characteristic property of linear isometries between two real normed planed in terms of antinorms, prove that a linear map T is a linear isometry if and only if the antinorm of Tx and x are equal , and that a linear map preseving Lumer orthogonality is a scalar multiple of the linear isometry.