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Let A and B be $ n\times n $ n × n positive semidefinite matrices, $ f:[0,\infty)\rightarrow \lbrack 0,\infty) $ f : [ 0 , ∞) → [ 0 , ∞) be an increasing convex function with $ f\left (0\right) =0 $ f (0) = 0 , and let $ \Phi : \mathbb {M}_{n}(\mathbb {C})\rightarrow \mathbb {M}_{n}(\mathbb {C}) $ Φ : M n (C) → M n (C) be a positive unital linear map. It is shown that $$\begin{align*} & \prod_{j=1}^{k} s_{j}\left(f^{4}\left(\Phi \left(\frac{A+B}{2}\right) \right) \right)\\ & \quad \leq 2^{-4} \prod_{j=1}^{k} \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(A\right) \right) +\Phi \left(f^{2}\left(A\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(B\right) \right) \right\Vert +1\right) \\ & \quad\quad \times \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(B\right) \right) +\Phi \left(f^{2}\left(B\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(A\right) \right) \right\Vert +1\right) \end{align*}$$ ∏ j = 1 k s j (f 4 (Φ (A + B 2))) ≤ 2 − 4 ∏ j = 1 k (s j (Φ 2 (f 2 (A)) + Φ (f 2 (A))) + ‖ Φ (f 2 (B)) ‖ + 1) × (s j (Φ 2 (f 2 (B)) + Φ (f 2 (B))) + ‖ Φ (f 2 (A)) ‖ + 1) for $ k=1,2,\ldots,n $ k = 1 , 2 , ... , n , where $ s_{j}\left (T\right) $ s j (T) and $ \left \Vert T\right \Vert $ ‖ T ‖ are the $ j{\rm th} $ j th singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given. [ABSTRACT FROM AUTHOR]

Let Mat n (F) denote the set of square n × n matrices over a field F of characteristic different from two. The permanental rank prk (A) of a matrix A ∈ Mat n (F) is the size of the maximal square submatrix in A with nonzero permanent. By Λ k and Λ ≤ k we denote the subsets of matrices A ∈ Mat n (F) with prk (A) = k and prk (A) ≤ k , respectively. In this paper for each 1 ≤ k ≤ n − 1 we obtain a complete characterization of linear maps T : Mat n (F) → Mat n (F) satisfying T (Λ ≤ k) = Λ ≤ k or bijective linear maps satisfying T (Λ ≤ k) ⊆ Λ ≤ k. Moreover, we show that if F is an infinite field, then Λ k is Zariski dense in Λ ≤ k and apply this to describe such bijective linear maps satisfying T (Λ k) ⊆ Λ k. [ABSTRACT FROM AUTHOR]

The multiplicative version of the Gleason–Kahane–Żelazko theorem for -algebras given by Brits et al. in [4] is extended to maps from -algebras to commutative semisimple Banach algebras. In particular, it is proved that if a multiplicative map from a -algebra to a commutative semisimple Banach algebra is continuous on the set of all noninvertible elements of and for any , then is a linear map. The multiplicative variation of the Kowalski–Słodkowski theorem given by Touré et al. in [14] is also generalized. Specifically, if is a continuous map from a -algebra to a commutative semisimple Banach algebra satisfying the conditions and for all , then generates a linear multiplicative map on which coincides with on the principal component of the invertible group of . If is a Banach algebra such that each element of has totally disconnected spectrum, then the map itself is linear and multiplicative on . It is shown that a similar statement is valid for a map with semisimple domain under a stricter spectral condition. Examples which demonstrate that some hypothesis in the results cannot be discarded. [ABSTRACT FROM AUTHOR]

The independence number of a square matrix A , denoted by α (A) , is the maximum order of its principal zero submatrices. Given any integer n , let S n + be the set of n × n nonnegative symmetric matrices with zero diagonal. We characterize the linear maps on S n + preserving the independence number of all matrices in S n +. [ABSTRACT FROM AUTHOR]

Let A and B be n × n positive semidefinite matrices, f : [ 0 , ∞ ) → [ 0 , ∞ ) be an increasing convex function with f ( 0 ) = 0 , and let Φ : M n ( C ) → M n ( C ) be a positive unital linear map. It is shown that ∏ j = 1 k s j ( f 4 ( Φ ( A + B 2 ) ) ) ≤ 2 − 4 ∏ j = 1 k ( s j ( Φ 2 ( f 2 ( A ) ) + Φ ( f 2 ( A ) ) ) + ‖ Φ ( f 2 ( B ) ) ‖ + 1 ) × ( s j ( Φ 2 ( f 2 ( B ) ) + Φ ( f 2 ( B ) ) ) + ‖ Φ ( f 2 ( A ) ) ‖ + 1 ) for k = 1 , 2 , … , n , where s j ( T ) and ‖ T ‖ are the j th singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given. [ABSTRACT FROM AUTHOR]

Let A⊂B(H) be a standard operator algebra on a Hilbert space H where dim H≥2, and A is closed under the adjoint operation. In this article, all linear maps δ,τ:A→B(H) satisfying Aτ(B)*+δ(A)B*=0(A*τ(B)+δ(A)*B=0) whenever AB*=0(A*B=0) are characterized, and as an application, linear maps on A behaving like right (left) centralizers at orthogonal elements are described. [ABSTRACT FROM AUTHOR]

In this paper, for a complex Hilbert space H with dimH = 2, we study the linear maps on B(H), the bounded linear operators on H, that preserves projections and idempotents. As a result, we characterize the linear maps on B(H) that preserves involutions in both directions. [ABSTRACT FROM AUTHOR]

We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree. [ABSTRACT FROM AUTHOR]

Let A be a unital standard algebra on a complex Banach space X with dim X ≥ 2 . The main result of this paper is to characterize the linear maps δ , τ : A → B (X) satisfying A τ (B) + δ (A) B = 0 whenever A , B ∈ A are such that A B = 0 . As application of our main result, we determine the linear map δ : A → B (H) that has one of the following properties for A , B ∈ A : if A B ⋆ = 0 , then A δ (B) ⋆ + δ (A) B ⋆ = 0 , or if A ⋆ B = 0 , then A ⋆ δ (B) + δ (A) ⋆ B = 0 , where A is a unital standard operator algebras on a Hilbert space H such that A is closed under the adjoint operation. We also provide other applications of the main result. [ABSTRACT FROM AUTHOR]

The decomposition of annihilator polynomials for linear maps or for the matrices of product of two coprime polynomials gives a decomposition of the spectrum as a direct sum of two subspaces with annihilator polynomials of smaller degree. [ABSTRACT FROM AUTHOR]

For a linear map T:V → V where V is a vector space, there are two special subspaces: the kernel (ker T) and the image (Im T =T(V)) with dimensions d(T) (the defect of T), respectively r(T) (the rank of T). In this paper we characterize some special linear maps (projections, symmetries and tripotent maps) using only these subspaces and their dimensions. [ABSTRACT FROM AUTHOR]

The definition of Choi matrices for linear maps on the n × n matrices is extended to factors, and the basic theorems for Choi matrices are proved in this more general context. [ABSTRACT FROM AUTHOR]

Let $$\mathcal {A}$$ be a unital Banach algebra and $$\mathcal {M}$$ be a unital Banach $$\mathcal {A}$$ -bimodule. The main results characterize a continuous linear map $$\varphi :\mathcal {A}\rightarrow \mathcal {M}$$ that satisfies $$a\varphi (a^{-1})=\varphi (1)$$ or $$a\varphi (a^{-1})+ \varphi (a^{-1}) a=2\varphi (1)$$ for all $$a$$ in principal component of invertible elements of $$\mathcal {A}$$ . The proof is based on the consideration of a continuous bilinear map satisfying a related condition. [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 X be a complex Banach space and let L(X) be the Banach algebra of all bounded linear operators on X. We characterize surjective linear maps ϕ : L(X) → L(X) compressing or depressing any one of the range, the hyper-range, the analytic core and the kernel. [ABSTRACT FROM AUTHOR]

Let 𝒜 be an algebra over a commutative unital ring 𝒞. We say that 𝒜 is zero product determined if for every 𝒞-module 𝒱 and every 𝒞-bilinear map φ: 𝒜 × 𝒜 → 𝒱 the following holds: if φ(A,B) = 0 wheneverAB = 0, then there exists a 𝒞-linear mapLsuch that φ(A,B) = L(AB) for allA,B ∈ 𝒜. If we replace, in this definition, the ordinary product by the Jordan (resp. Lie) product, then we say that 𝒜 is zero Jordan (resp. Lie) product determined. We show that the triangular algebrais zero (resp. Lie) product determined if and only if 𝒜 and ℬ are zero (resp. Lie) product determined, and under some technical restrictions, a same result is true for the Jordan product. The main result is then applied to generalized triangular matrix algebras and (block) upper triangular matrix algebras. In particular, we show that: (i) the block upper triangular matrix algebra(n ≥ 1) is a zero product determined algebra; (ii) if 𝒞 contains the element, then(n ≥ 1) is a zero Jordan product determined algebra and (iii) if 𝒜 is a commutative algebra, then(n ≥ 1) is a zero Lie product determined algebra. [ABSTRACT FROM AUTHOR]

Abstract: Let Alg be a nest algebra associated with the nest on a (real or complex) Hilbert space . We say that Alg is zero product determined if for every linear space and every bilinear map the following holds: if ϕ(A,B)=0 whenever AB =0, then there exists a linear map T such that for all . If we replace in this definition the ordinary product by the Jordan (resp., Lie) product, then we say that Alg is zero Jordan (resp., Lie) product determined. We show that any finite nest algebra over a complex Hilbert space is zero product determined, and it is also zero Jordan product determined. Moreover, we show that any finite-dimensional nest algebra on a (real or complex) Hilbert space is zero Lie (resp., associative, Jordan) product determined. In addition, we characterize separately strongly operator topology continuous bilinear map ϕ from Alg × Alg into a topological linear space with the property that whenever AB =0 or whenever AB + BA =0. [Copyright &y& Elsevier]

Abstract: In this paper we fully describe all tropical linear maps in the tropical projective plane , that is, maps from the tropical plane to itself given by tropical multiplication by a real matrix A. The map is continuous and piecewise-linear in the classical sense. In some particular cases, the map is a parallel projection onto the set spanned by the columns of A. In the general case, after a change of coordinates, the map collapses at most three regions of the plane onto certain segments, called antennas, and is a parallel projection elsewhere (Theorem 3). In order to study , we may assume that A is normal, i.e., , up to changes of coordinates. A given matrix A admits infinitely many normalizations. Our approach is to define and compute a unique normalization for A (which we call lower canonical normalization) (Theorem 1) and then always work with it, due both to its algebraic simplicity and its geometrical meaning. On , any , some aspects of tropical linear maps have been studied in [6]. We work in , adding a geometric view and doing everything explicitly. We give precise pictures. Inspiration for this paper comes from [3,5,6,8,12,26]. We have tried to make it self-contained. Our preparatory results present noticeable relationships between the algebraic properties of a given matrix A (idempotent normal matrix, permutation matrix, etc.) and classical geometric properties of the points spanned by the columns of A (classical convexity and others); see Theorem 2 and Corollary 1. As a by-product, we compute all the tropical square roots of normal matrices of a certain type; see Corollary 4. This is, perhaps, a curious result in tropical algebra. Our final aim is, however, to give a precise description of the map . This is particularly easy when two tropical triangles arising from A (denoted and ) fit as much as possible. Then the action of is easily described on (the closure of) each cell of the cell decomposition ; see Theorem 3. Normal matrices play a crucial role in this paper. The tropical powers of normal matrices of size satisfy . This statement can be traced back, at least, to [26], and appears later many times, such as [1,2,6,9,10]. In lemma 1, we give a direct proof of this fact, for . But now the equality means that the columns of are three fixed points of and, in fact, any point spanned by the columns of is fixed by . Among normal matrices, the idempotent ones (i.e., those satisfying) are particularly nice: we prove that the columns of such a matrix tropically span a set which is classically compact, connected and convex (Lemma 2 and Corollary 1). In our terminology, it is a good tropical triangle. [Copyright &y& Elsevier]

We show that the Jordan algebra S of symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that if a bilinear map {., .} from S × S into a vector space X satisfies {x, y} = 0 whenever x ○ y = 0, then there exists a linear map T : S → X such that {x, y} = T(x ○ y) for all x, y ∈ S (here, x ○ y = xy + yx). [ABSTRACT FROM AUTHOR]

Abstract: Let be a vector space over a field . Assume that the characteristic of is large, i.e. . Let be an invertible linear map. We answer the following question in this paper. When does admit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question. Following Feit and Zuckerman , an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in . As a consequence of the answers to the above question, we offer a characterization of the real elements in . Suppose is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group . A non-degenerate S-invariant subspace of is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in . The level of a unipotent T is the least integer k such that . We also classify the levels of unipotents in . [ABSTRACT FROM AUTHOR]

We show that the Lie algebra L of skew-symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that every bilinear map {·,·} from L × L into a vector space X is of the form {x, y} = T ([x, y]) for some linear map T provided that the following condition is fulfilled: [x, y] = 0 implies {x, y} = 0. [ABSTRACT FROM AUTHOR]

A V-multiplier is a self-map f on a V-semilattice L satisfying [image omitted][image omitted] for every [image omitted]. We introduce some properties of V-multiplier and define expansive linear and filtered expansive linear maps on semilattices (Λ-semilattices) and research the properties of the expansive linear maps, and then we prove that L is a modular lattice if and only if [image omitted], where [image omitted] is the family of all filtered expansive linear maps and [image omitted] the family of all V-multipliers on L, and that [image omitted] and [image omitted] is embedded in the complete lattice [image omitted] of the all filters of L. [ABSTRACT FROM AUTHOR]

Abstract: Let , , be the algebra of all matrices over a field of characteristic not , and let be a bilinear map from into an arbitrary vector space over . Our main result states that if whenever and are orthogonal rank one idempotents, then there exist linear maps such that for all . This is applicable to some linear preserver problems. [Copyright &y& Elsevier]

Abstract: Let A be an algebra over a commutative unital ring C. We say that A is zero product determined if for every C-module X and every bilinear map the following holds: if whenever , then there exists a linear operator T such that for all . If we replace in this definition the ordinary product by the Lie (resp. Jordan) product, then we say that A is zero Lie (resp. Jordan) product determined. We show that the matrix algebra , , where B is any unital algebra, is always zero product determined, and under some technical restrictions it is also zero Jordan product determined. The bulk of the paper is devoted to the problem whether is zero Lie product determined. We show that this does not hold true for all unital algebras B. However, if B is zero Lie product determined, then so is . [Copyright &y& Elsevier]

Abstract: The main result of the paper is the following: If an F-space X is covered by a family of sets such that whenever , and f is a linear map from X to a topological linear space Y which is continuous on each of the sets , then f is continuous. This provides a very strong negative answer to a problem posed recently by J. Ka̧kol and M. López Pellicer. A number of consequences of this result are given, some of which are quite curious. Also, inspired by a related question asked by J. Ka̧kol, it is shown that if a linear map is continuous on each member of a sequence of compact sets, then it is also continuous on every compact convex set contained in the linear span of the sequence. The construction applied to prove this is then used to interpret a natural linear topology associated with the sequence as the inductive limit topology in the sense of Ph. Turpin, and thus derive its basic properties. [Copyright &y& Elsevier]

Abstract: In this note we consider linear maps on the complex n × n matrix algebra preserving products of either two positive matrices, a positive matrix and a Hermitian matrix, or two Hermitian matrices. We give characterizations of all those injections. [Copyright &y& Elsevier]

Given a nilpotent endomorphism, we consider the manifold of invariant subspaces having a fixed Segre characteristic. In [Linear Algebra Appl., 332–334 (2001) 569], the implicit form of a miniversal deformation of an invariant subspace with respect to the usual equivalence relation between subspaces is obtained. Here we obtain the explicit form of this deformation when the invariant subspace is marked, and we use it to calculate the dimension of the orbit and in particular to characterize the stable marked subspaces (those with open orbit). Moreover, we study the rank of the endomorphisms in the quotient space by the subspaces in the miniversal deformation of the giving subspace. [Copyright &y& Elsevier]

There are many characterizations of linear operators from various matrix spaces into themselves which preserve term rank. In this research, we characterize the linear maps which preserve any two term ranks between different matrix spaces over anti-negative semirings, which extends the previous results on characterizations of linear operators from some matrix spaces into themselves. That is, a linear map T from p × q matrix spaces into m × n matrix spaces preserves any two term ranks if and only if T preserves all term ranks if and only if T is a ( P , Q , B )-block map. [ABSTRACT FROM AUTHOR]