Linear maps preserving Ky Fan norms and Schatten norms of tensor products of matrices

الأصل · EN

For a positive integer n, let Mₙ be the set of n× n complex matrices. Suppose · is the Ky Fan k-norm with 1 ≤ k ≤ mn or the Schatten p-norm with 1 ≤ p ≤ ∞ (p≠ 2) on Mmn, where m,n≥ 2 are positive integers. It is shown that a linear map ϕ: Mmn → Mmn satisfying A⊗ B = ϕ(A⊗ B) for all A ∈ Mₘ and B ∈ Mₙ if and only if there are unitary U, V ∈ Mmn such that ϕ has the form A⊗ B U(φ₁(A) ⊗ φ₂(B))V, where φₛ(X) is either the identity map X X or the transposition map X Xᵗ. The results are extended to tensor space Mₙ₁ ⊗ ⊗ Mₙₘ of higher level. The connection of the problem to quantum information science is mentioned.