Linear Maps Preserving Orthogonality

To investigate the geometry of normed spaces, many orthogonality relations aredefined. For example, a vector x in a normed space X is said to be orthogonal in theBirkhof sence to a vector y in X, if for every scalar α we have x+αy≥x. Koldob-sky proved that a linear map between real normed spaces which preserves Birkhoforthogonality must be a scalar multiple of an isometry. Blanco and Turnˇsek extendedthis result to the complex setting. For x, y∈X, let ρ2±(x, y):=limx+ty2xt→0±2t=We call that a vector x is ρ+(ρ) orthogonality to ρ+(x, y)=0(ρ (x, y)=0). Also call that a vector x is ρ orthogonality to a vector y, ifρ+(x, y)+ρ (x, y)=0. Chmielin′ski and W′ojcik proved that a linear map between realnormed spaces which preserves ρ+orthogonality and ρ orthogonality must be a scalarmultiple of an isometry. Wo′jcik proved that a linear map between real normed spaceswhich preserves ρ orthogonality must be a scalar multiple of an isometry. The mainpurpose in this article is to generalize the above results to complex normed spaces.(1) We consider the properties of ρ+orthogonality and ρ orthogonality and prove thata linear map between complex normed spaces that preserves ρ+orthogonality and ρ orthogonality must be a scalar multiple of an isometry.(2) We define a new orthogonality and prove that a linear map between complex normedspaces that preserves this orthogonality relation must be a scalar multiple of an isom-etry.(3) We prove that a linear map between complex normed spaces that preserves ρ or-thogonality must be a scalar multiple of an isometry.