2024/10/10 by Li, Lei, Liu, Siyu, Peralta, Antonio M.
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2410.08101
Let H and K be two complex inner product spaces with dim(X)≥ 2. We prove that for each non-zero additive mapping A:H → K with dense image the following statements are equivalent: (a) A is (complex) linear or conjugate-linear mapping and there exists γ>0 such that ‖ A (x) ‖ = γ‖x‖, for all x∈ X, that is, A is a positive scalar multiple of a linear or a conjugate-linear isometry; (b) There exists γ1 >0 such that one of the next properties holds for all x,y ∈ H: (b.1) ⟨ A(x) |A(y)⟩ = γ1 ⟨ x|y⟩, (b.2) ⟨ A(x) |A(y)⟩ = γ1 ⟨ y|x ⟩; (c) A is linear or conjugate-linear and preserves orthogonality in both directions; (d) A is linear or conjugate-linear and preserves orthogonality; (e) A is additive and preserves orthogonality in both directions; (f) A is additive and preserves orthogonality. This extends to the complex setting a recent generalization of the Koldobsky--Blanco--Turnšek theorem obtained by Wójcik for real normed spaces.