Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products

Suppose m, n ≥ 2 are positive integers. Let 𝒯n be the space of all n × n complex upper triangular matrices, and let ϕ be an injective linear map on 𝒯m ⊗ 𝒯n. Then ϕ(A ⊗ B) is an idempotent matrix in 𝒯m ⊗ 𝒯n whenever A ⊗ B is an idempotent matrix in 𝒯m ⊗ 𝒯n if and only if there exists an invertible matrix P ∈ 𝒯m ⊗ 𝒯n such that ϕ(A ⊗ B) = P(ξ1(A) ⊗ ξ2(B))P−1, ∀A ∈ 𝒯m, B ∈ 𝒯n, or when m = n, ϕ(A ⊗ B) = P(ξ1(B) ⊗ ξ2(A))P−1, ∀A ∈ 𝒯m, B ∈ 𝒯m, where ξ1([aij]) = [aij] or ξ1([aij]) = [am−i+1, m−j+1] and ξ2([bij]) = [bij] or ξ2([bij]) = [bn−i+1, n−j+1].