Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products

الملخص EN

Suppose m , n ≥ 2 are positive integers.

Let ? n be the space of all n × n complex upper triangular matrices, and let ϕ be an injective linear map on ? m ⊗ ? n .

Then ϕ ( A ⊗ B ) is an idempotent matrix in ? m ⊗ ? n whenever A ⊗ B is an idempotent matrix in ? m ⊗ ? n if and only if there exists an invertible matrix P ∈ ? m ⊗ ? n such that ϕ ( A ⊗ B ) = P ( ξ 1 ( A ) ⊗ ξ 2 ( B ) ) P - 1 , ∀ A ∈ ? m , B ∈ ? n , or when m = n , ϕ ( A ⊗ B ) = P ( ξ 1 ( B ) ⊗ ξ 2 ( A ) ) P - 1 , ∀ A ∈ ? m , B ∈ ? m , where ξ 1 ( [ a i j ] ) = [ a i j ] or ξ 1 ( [ a i j ] ) = [ a m - i + 1 , m - j + 1 ] and ξ 2 ( [ b i j ] ) = [ b i j ] or ξ 2 ( [ b i j ] ) = [ b n - i + 1 , n - j + 1 ] .