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A map Φ between matrices is said to be zero product preserving if $$\begin{align*} \Phi(A)\Phi(B) = 0 \quad {\rm whenever}\ AB = 0. \end{align*}$$ Φ (A) Φ (B) = 0 whenever AB = 0. In this paper, we give concrete descriptions of an additive/linear zero product preserver $ \Phi : \mathbf{M}_n(\mathbb {F}) \rightarrow \mathbf{M}_r(\mathbb {F}) $ Φ : M n (F) → M r (F) between matrix algebras of different dimensions over an arbitrary field $ \mathbb {F} $ F , and $ n\geq 2 $ n ≥ 2. In particular, we show that if Φ is linear and preserves zero products then $$\begin{align*} \Phi(A)= S\begin{pmatrix} R_1 \otimes A & 0 \\ 0 & \Phi_0(A)\end{pmatrix} S^{-1}, \end{align*}$$ Φ (A) = S ( R 1 ⊗ A 0 0 Φ 0 (A) ) S − 1 , for some invertible matrices $ R_1 $ R 1 in $ \mathbf{M}_k(\mathbb {F}) $ M k (F) , S in $ \mathbf{M}_r(\mathbb {F}) $ M r (F) and a zero product preserving linear map $ \Phi _0: \mathbf{M}_n(\mathbb {F}) \rightarrow \mathbf{M}_{r-nk}(\mathbb {F}) $ Φ 0 : M n (F) → M r − nk (F) into nilpotent matrices. If $ \Phi (I_n) $ Φ (I n) is invertible, then $ \Phi _0 $ Φ 0 is vacuous. In general, the structure of $ \Phi _0 $ Φ 0 could be quite arbitrary, especially when $ \Phi _0(\mathbf{M}_n({\mathbb F})) $ Φ 0 (M n (F)) has trivial multiplication, i.e. $ \Phi _0(X)\Phi _0(Y) = 0 $ Φ 0 (X) Φ 0 (Y) = 0 for all X, Y in $ \mathbf{M}_n(\mathbb {F}) $ M n (F). We show that if $ \Phi _0(I_n) = 0 $ Φ 0 (I n) = 0 or $ r-nk \le n+1 $ r − nk ≤ n + 1 , then $ \Phi _0(\mathbf{M}_n({\mathbb F})) $ Φ 0 (M n (F)) indeed has trivial multiplication. More generally, we characterize subspaces $ \mathbf{V} $ V of square matrices satisfying XY = 0 for any $ X, Y \in \mathbf{V} $ X , Y ∈ V . Similar results for double zero product preserving maps are obtained. [ABSTRACT FROM AUTHOR]

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]

Since the introduction of majorization in R n by Hardy, Littlewood and Polya, several extensions of this concept have been studied in the literature. Recently, Kosuru and Saha [Cone type majorization and its strong linear preservers. Electron J Linear Algebra. 2020;36(36):511–518.] defined the concept of cone type majorization. In this paper, we focus on the study of the behavior of the linear preservers of majorization and cone type majorization under generalized inversion, namely, Drazin inversion and Moore-Penrose inversion. A characterization of these linear preservers, given by Ando [Majorization, doubly stochastic matrices and comparisons of eigenvalues. Linear Algebra Appl. 1989;118:163–248.] for majorization, and by Kosuru and Saha [Cone type majorization and its strong linear preservers. Electron J Linear Algebra. 2020;36(36):511–518.] for cone type majorization, prove to be crucial in our proofs. [ABSTRACT FROM AUTHOR]

A positive answer on a question posed in Eshkaftaki et al. [DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] is provided. Precisely, the concrete form of bounded linear operators which preserve DSS-weak majorization on discrete Lebesgue space ℓ 1 (I) , when I is an infinite set, is presented. Interestingly, it is shown that the class of DSS-weak majorization preservers is a proper subset of the class of majorization preservers on ℓ 1 (I) , despite the fact that these two classes of preservers coincide [Eshkaftaki et al. DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] when above relations are defined on ℓ p (I) when p ∈ (1 , ∞). In this way, the conjecture given at the end of the paper [Eshkaftaki et al. DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] is confirmed. [ABSTRACT FROM AUTHOR]

Let X be a real or complex Banach space with dimension at least 3. Denote by B (X) the Banach algebra of all bounded linear operators on X. In this paper, we characterize surjective maps Φ : B (X) → B (X) satisfying A + B ∼ C if and only if Φ (A) + Φ (B) ∼ Φ (C) for all A , B , C ∈ B (X). [ABSTRACT FROM AUTHOR]

In this paper, we present results concerning orthogonality in Hilbert C ∗ -modules. Moreover, for a C ∗ -algebra A , we prove theorems concerning the multi- A -linearity and its preservation by A -linear operators. New version of solution of the orthogonality equation on Hilbert C ∗ -modules and mappings preserving orthogonality are also investigated. [ABSTRACT FROM AUTHOR]

In the present paper, we will characterize linear maps of M n (F) , the ring of all n × n matrices over an algebraically closed field F of characteristic zero, which preserve the set of all algebraic elements of degree 2. [ABSTRACT FROM AUTHOR]

Let G L n be a complex general linear group. In this paper we determine the structure of all Jordan triple product maps from G L 1 to G L m for m = 2 , 3. The results are applied to determine the structure of all continuous Jordan triple product maps between G L 1 and G L m for m = 2 , 3. These are the continuous maps ϕ : G L 1 → G L m which satisfy ϕ (z w z) = ϕ (z) ϕ (w) ϕ (z) , z , w ∈ G L 1. [ABSTRACT FROM AUTHOR]

In this paper, we give the characterization of Pareto eigenvalues preserving linear maps on real square matrices. In fact, we show that a linear mapon, the algebra of allnbynmatrices with real entries, preserves Pareto eigenvalues if and only if, for some nonnegative generalized permutation matrixP. [ABSTRACT FROM PUBLISHER]

The commutation matrix was first introduced in statistics as a transposition matrix by Murnaghan in 1938. In this paper, we first investigate the commutation matrix which is employed to transform a matrix into its transpose. We then extend the concept of the commutation matrix to commutation tensor and use the commutation tensor to achieve the unification of the two formulae of the linear preserver of the matrix rank, a classical result of Marcus in 1971. [ABSTRACT FROM AUTHOR]

Let X and Y be real matrices. We say that X is circulant matrix majorized by Y and write , if for some circulant doubly stochastic matrix. A linear operator is said to be a preserver of , if whenever for. The main purpose of this paper is to characterize linear preservers of on real rectangular matrices. Moreover, we state a simple proof for characterization of the linear preservers of circulant majorization on [ABSTRACT FROM AUTHOR]

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]

This paper concerns two notions of matrix functions over semirings; zero-term rank and zero-star cover number. We study linear operators that preserve these two matrix functions ofsymmetric matrices. Consequently, we determine the form of linear operatorsTonsymmetric matrices that preserves zero-term rank (zero-star cover number). We also show that a linear operatorTon symmetric matrices preserves zero-term rank (zero-star cover number) if and only ifTpreserves zero-term ranks (zero-star cover numbers, respectively) 1 and 2. Other characterizations of zero-term rank (zero-star cover number, respectively) preservers are also given. [ABSTRACT FROM PUBLISHER]

Let G L n be a complex general linear group. In this paper we determine the structure of all Jordan triple product maps from G L 1 to G L m for m = 2 , 3. The results are applied to determine the structure of all continuous Jordan triple product maps between G L 1 and G L m for m = 2 , 3. These are the continuous maps ϕ : G L 1 → G L m which satisfy ϕ (z w z) = ϕ (z) ϕ (w) ϕ (z) , z , w ∈ G L 1. [ABSTRACT FROM AUTHOR]

Letbe a division ring withand, where, for some positive integer. A vectoris called an eigenvector ofif there existssuch that. In this paper, we characterize the additive mapsuch that every eigenvector ofis again an eigenvector offor all. [ABSTRACT FROM PUBLISHER]

The goal of the present paper is to derive some conditions on saturation of (strong) subadditivity inequality for the stochastic matrices. The notion of relative entropy of stochastic matrices is introduced by mimicking quantum relative entropy. Some properties of this concept are listed, and the connection between the entropy of the stochastic quantum operations and that of stochastic matrices are discussed. [ABSTRACT FROM AUTHOR]

Letbe the space of allupper Hessenberg matrices over a fieldwhereis a positive integer greater than. In this paper, linear maps preserving rank-onare characterized, i.e.is a linear rank-preserver onif and only if (i)is an-dimensional rank-subspace or (ii) there exist nonsingular upper Hessenberg matricesandsuch thatfor allorfor allwherewith. [ABSTRACT FROM AUTHOR]

Let be an arbitrary linear space and an n-linear map. It is proved that, for each choice of a basis of , the n-linear map f induces a (nontrivial) decomposition as a direct sum of linear subspaces of , with respect to. It is shown that this decomposition is f-orthogonal in the sense that when , and in such a way that any is strongly f-invariant, meaning that A sufficient condition for two different decompositions of induced by an n-linear map f, with respect to two different bases of , being isomorphic is deduced. The f-simplicity – an analog of the usual simplicity in the framework of n-linear maps – of any linear subspace of a certain decomposition induced by f is characterized. Finally, an application to the structure theory of arbitrary n-ary algebras is provided. This work is a close generalization the results obtained by Calderón. [ABSTRACT FROM AUTHOR]

We introduce an equivalence relation ρ on matrices over a class of semirings, which is relevant to the Kleene star operator on matrices. We characterize those surjective linear transformations for matrices preserving ρ. Also, we show that a surjective linear transformation preserves Kleene stars of matrices if and only if it commutes with the Kleene star operator if and only if it preserves ρ. Finally, we illustrate that there exist non-surjective linear transformations that preserve ρ or preserve Kleene stars of matrices or commute with the Kleene star operator. [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 n>1 be a positive integer, {H1, ..., Hn} be a finite collection of complex Hilbert spaces with dim ⁡ (H k) ≥ 2 , and P1(Hk) be the set of all rank-1 self-adjoint projections on Hk, k = 1, ..., n. Set D P 1 ⨂ k = 1 n H k = { A 1 ⊗ ⋯ ⊗ A n : A k ∈ P 1 (H k) , k = 1 , ... , n }. We characterize the maps ϕ from D (P 1 (⨂ k = 1 n H k)) to P 1 (⨂ k = 1 n H k) preserving transition probability, i.e. t r (A B) = t r (ϕ (A) ϕ (B)) , f o r a l l A , B ∈ D P 1 ⨂ k = 1 n H k . A particular case corresponding to n = 1 is well known as (non-surjective version) Wigner's theorem. Our result may be considered as a generalization of Wigner's theorem. [ABSTRACT FROM AUTHOR]

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose i j t h entry (for i ≠ j) is nonzero whenever (i , j) is an edge in G and is zero otherwise. The sum of the minimum rank of a graph and its maximum nullity (similarly defined) is always the number of vertices in G. This article compares the minimum rank with the clique covering number of G and the Boolean rank of its adjacency matrix. It does the same analysis for bipartite graphs. Finally, we investigate the linear operators on the set of graphs on n vertices that preserve the minimum rank. [ABSTRACT FROM AUTHOR]

A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k−1. A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and that maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus not k to graphs of genus not k. We show that such linear operators are necessarily vertex permutations. [ABSTRACT FROM AUTHOR]

In various normed spaces we answer the question of when a given isometry is a square of some isometry. In particular, we consider (real and complex) matrix spaces equipped with unitarily invariant norms and unitary congruence invariant norms, as well as some infinite dimensional spaces illustrating the difference between finite and infinite dimensions. [ABSTRACT FROM AUTHOR]

We characterize continuous, nondegenerate multiplicative maps on the algebraofupper-triangular Toeplitz matrices. As an application, we show that the continuous multiplicative isometries of(for any norm) are necessarily similarities or similarities composed with complex conjugation; for unitarily invariant norms, such isometries are precisely the unitary similarities, or unitary similarities composed with complex conjugation. We partially extend these results to algebras generated by non-derogatory matrices. [ABSTRACT FROM PUBLISHER]

Letdenote the set ofsymmetric matrices over some semiring. The set,, of elements ofthat are a finite sum of squares of elements in, is called the positive subsemiring of. A matrix,B, with entries inis primitive if some power ofBhas all nonzero entries. We characterize those linear operators on the set of symmetric matrices overthat map the set of primitive matrices to itself and the set of nonprimitive matrices to itself. We also characterize those linear operators on the set of symmetric matrices overthat map the set of primitive matrices whose square has all nonzero entries to itself and the complement of that set to itself. The characterization of linear preservers of real symmetric primitive matrices follows as a special case. [ABSTRACT FROM AUTHOR]

Letdenote the set ofmatrices with entries in. We writeto denote the subset of, all of whose entries are non-negative. Letdenote the set of allrealsymmetricmatrices. A matrixis said to becompletely positiveif there is some matrixsuch that. TheCP-rankof the matrixis the smallestsuch thatfor some. In this article, we shall investigate the linear operators onthat preserve sets of matrices defined by the CP-rank. We classify those that preserve the CP-rank function, those that preserve the set of CP-rank-1 matrices, those that preserve the sets of CP-rank-1 matrices and the set of CP-rank-2 matrices, and those that strongly preserve the set of CP-rank-1 matrices. [ABSTRACT FROM PUBLISHER]

In this note, we characterize linear maps between spaces of Hermitian matrices over a field with involution which preserve the rank of the identity matrix and the rank of tensor products of rank one Hermitian matrices. It is assumed that the fixed field with respect to the field with involution has at least four elements. We also present some applications of our main result to other preserver problems. [ABSTRACT FROM AUTHOR]

We characterize surjective maps between two unital-algebras which compress and expand the numerical range of the product and triple product of elements in such-algebras. [ABSTRACT FROM PUBLISHER]

Letbe a generalized matrix algebra over a commutative ringandbe the centre of. Suppose thatis an-bilinear mapping andis the trace of. We describe the form ofsatisfying the conditionfor all. The question of whenhas the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism ofto be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article. [ABSTRACT FROM PUBLISHER]

We consider three linear preserver problems on the algebra of infinite triangular matrices over fields. We characterize the maps preserving invertible and noninvertible matrices, the surjective maps preserving inverses and the surjective maps preserving rank permutability. [ABSTRACT FROM AUTHOR]

Letbe the algebra of allcomplex matrices. We characterize linear mapssatisfyingfor all,. [ABSTRACT FROM AUTHOR]

For a positive integer, letbe the set ofcomplex matrices. Supposeare positive integers and. Denote bythe-numerical range of a matrix. It is shown that a linear mapsatisfiesfor allandif and only if there is a unitarysuch that one of the following holds.   For all , ,     and for all , ,where (1)is the identity mapor the transposition map, or (2)andhas the formor. [ABSTRACT FROM PUBLISHER]

Letdenote the two element Boolean algebra andbe the semimodule of alln-square symmetric Boolean matrices with zero diagonal. We characterize (i) subsemimodules ofwhose nonzero members all have factor rank 2, (ii) linear mappings fromtothat send distinct elements of term rank 2 to distinct elements of factor rank 2 and (iii) linear mappings fromtothat preserve elements of factor rank 2 and also elements of factor rankkfor some [ABSTRACT FROM PUBLISHER]

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ onmn × mnHermitian matrices such that φ(A ⊗ B) andA ⊗ Bhave the same spectrum for anym × mHermitianAandn × nHermitianB. Such a map has the formA ⊗ B ↦ U(ϕ1(A) ⊗ ϕ2(B))U* formn × mnHermitian matrices in tensor formA ⊗ B, whereUis a unitary matrix, and forj ∈ {1, 2}, ϕjis the identity map X ↦ Xor the transposition map X ↦ Xt. The structure of linear maps leaving invariant the spectral radius of matrices in tensor formA ⊗ Bis also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems. [ABSTRACT FROM AUTHOR]

ForA,C ∈ MntheC-determinantal range ofAis the following set on the complex plane ▵C(A) = {det(A − UCU*):UU* = In}. For normal matricesAandCwith eigenvalues α1, … , αnand γ1, … , γn, respectively, Marcus [M. Marcus,Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J.22(1973), pp. 1137–1149] and Oliveira [G.N. de Oliveira,Normal matrices (research problem), Linear Multilinear Algebra12(1982), pp. 153–154] conjectured that ▵C(A) is a subset of the convex hull of the points, σ ∈ Sn, whereSnis the symmetric group of degreen. We investigate the extremal set of matrices for which the equality holds in the Marcus–Oliveira conjecture. We illustrate the use of the obtained results by two different applications. The first one deals with the equality case between the radius of ▵C(A) and the radius of the convex hull of the pointszσ, σ ∈ Sn. The second one is the characterization of additive Frobenius endomorphisms for the determinantal range or radius on the spaceMnand on its real subspace of Hermitian matrices. [ABSTRACT FROM PUBLISHER]

We investigate bijective maps of unitriangular matrices which preserve commutators. We show that every such map is a composition of at most four standard maps preserving commutators, of which three are automorphisms. Next, using this result, we obtain an analogous description for maps of triangular matrices preserving commutators. We prove that they are compositions of at most three standard maps, of which two are automorphisms. [ABSTRACT FROM AUTHOR]