Search

Let $ n\geqslant 2 $ n ⩾ 2 be an integer and let $ T_n({\mathbb {D}}) $ T n (D) be the ring of $ n\times n $ n × n upper triangular matrices over a division ring $ {\mathbb {D}} $ D with centre $ Z(T_n({\mathbb {D}})) $ Z (T n (D)). In this paper, we characterize additive maps $ \psi :T_n({\mathbb {D}})\rightarrow T_n({\mathbb {D}}) $ ψ : T n (D) → T n (D) satisfying $ [\psi (A),\psi (B)]-[A,B]\in Z(T_n({\mathbb {D}})) $ [ ψ (A) , ψ (B) ] − [ A , B ] ∈ Z (T n (D)) for all $ A,B\in T_n({\mathbb {D}}) $ A , B ∈ T n (D). We then deduce from this result a complete characterization of strong commutativity preserving additive maps $ \psi :T_n({\mathbb {D}})\rightarrow T_n({\mathbb {D}}) $ ψ : T n (D) → T n (D) on rank k upper triangular matrices, where $ 1\leqslant k\leqslant n $ 1 ⩽ k ⩽ n is a fixed integer such that $ k\neq n $ k ≠ n when $ |{\mathbb {D}}|=2 $ | D | = 2. [ABSTRACT FROM AUTHOR]

Let k , n 1 , ... , n k be positive integers such that n i ⩾ 2 for i = 1 , ... , k and let M n i denote the algebra of n i × n i matrices over a field F for i = 1 , ... , k. Let ⨂ i = 1 k M n i be the tensor product of M n 1 , ... , M n k . We obtain a structural characterization of additive maps ψ : ⨂ i = 1 k M n i → ⨂ i = 1 k M n i satisfying ψ ⨂ i = 1 k A i ⨂ i = 1 k A i = ⨂ i = 1 k A i ψ ⨂ i = 1 k A i for all A 1 ∈ S n 1 , ... , A k ∈ S n k , where S n i = E s t (n i) + α E p q (n i) : α ∈ F , 1 ⩽ p , q , s , t ⩽ n i a r e n o t a l l d i s t i n c t i n t e g e r s E p q (n i) and E s t (n i) is the standard matrix unit in M n i for i = 1 , ... , k. In particular, we show that ψ : M n 1 → M n 1 is an additive map commuting on S n 1 if and only if there exist a scalar λ ∈ F and an additive map μ : M n 1 → F such that ψ (A) = λ A + μ (A) I n 1 for all A ∈ M n 1 . As an application, we classify additive maps ψ : ⨂ i = 1 k M n i → ⨂ i = 1 k M n i satisfying ψ (⨂ i = 1 k A i) (⨂ i = 1 k A i) = (⨂ i = 1 k A i) ψ (⨂ i = 1 k A i) for all A 1 ∈ R r 1 n 1 , ... , A k ∈ R r k n k . Here, R r i n i denotes the set of rank r i matrices in M n i and each 1 < r i ⩽ n i is a fixed integer such that r i ≠ n i when n i = 2 and | F | = 2 for i = 1 , ... , k. [ABSTRACT FROM AUTHOR]

Let Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i be a linear map on the Kronecker product of spaces of Hermitian matrices H n i of size n i ≥ 3. (If d=1, we identify ⨂ i = 1 d H n i with H n 1 .) We establish a condition under which Ψ adj ⨂ i = 1 d A i = adj Ψ ⨂ i = 1 d A i if and only if det Ψ ⨂ i = 1 d A i = det ⨂ i = 1 d A i for all ⨂ i = 1 d A i ∈ ⨂ i = 1 d H n i . Then for d ∈ { 1 , 2 } , we apply this fact to characterize maps Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i such that Ψ adj ⨂ i = 1 d A i = adj Ψ ⨂ i = 1 d A i ⨂ i = 1 d with some mild conditions. [ABSTRACT FROM AUTHOR]

Let n ⩾ 2 be an integer and let F be a field with F ⩾ 3. Let T n (F) be the ring of n × n upper triangular matrices over F with centre Z . Fixing an integer 2 ⩽ k ⩽ n , we prove that an additive map ψ : T n (F) → T n (F) satisfies A ψ (A) = ψ (A) A for all rank k matrices A ∈ T n (F) if and only if there exist an additive map μ : T n (F) → Z , Z ∈ Z and α ∈ F in which α = 0 when F > 3 or k

Let and be fields and let n be a positive integer. Let and be linear spaces over such that and let and be linear spaces over. Let be the tensor product of and and let be the subspace of symmetric tensors of. In this paper, we show that a map satisfies for all rank r tensors if and only if is additive. Here r is a fixed integer such that and with , or with and. Examples showing the indispensability of assumption in our results are included. [ABSTRACT FROM AUTHOR]

Letbe the algebra ofmatrices over a field with at least three elements. Inspired by linear preserver problems related to quantum information science, we characterize classical adjoint commuting linear maps, i.e. linear mapssuch that with, that preserve decomposable tensors of. Some examples are given to distinguish the study of classical adjoint commuting linear maps of tensor products of matrices from those found in the study of classical adjoint commuting linear maps on matrix spaces. [ABSTRACT FROM AUTHOR]