Strong commutativity preserving maps on Lie ideals

Let A be a prime ring and let R be a noncentral Lie ideal of A. An additive map f:R→A is called strong commutativity preserving (SCP) on R if [f(x),f(y)]=[x,y] for all x,y∈R. In this paper we show that if f is SCP on R, then there exist λ∈C, λ²=1 and an additive map μ:R→Z(A) such that f(x)=λx+μ(x) for all x∈R where C is the extended centroid of A, unless charA=2 and A satisfies the standard identity of degree 4.     

Commuting traces and commutativity preserving maps on triangular algebras

Let A be a triangular algebra. The problem of describing the form of a bilinear map satisfying B(x,x)x=xB(x,x) for all xA is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.     

Strong commutativity preserving maps in prime rings with involution

Let A be a prime ring of characteristic not 2, with center Z(A) and with involution *. Let S be the set of symmetric elements of A. Suppose that f:S→A is an additive map such that [f(x),f(y)]=[x,y] for all x,y∈S. Then unless A is an order in a 4-dimensional central simple algebra, there exists an additive map μ:S→Z(A) such that f(x)=x+μ(x) for all x∈S or f(x)=-x+μ(x) for all x∈S.     

Linear maps preserving reduced norms

If a linear map between central simple algebras preserves reduced norms, it is an isomorphism or antiisomorphism followed by multiplication by an element of reduced norm 1.     

Linear maps preserving group majorization

A necessary and sufficient condition for a linear map to preserve group majorizations is given. The condition is applied to prove some preservation results.     

The group of commutativity preserving maps on strictly upper triangular matrices

Let N = N _n (R) be the algebra of all n × n strictly upper triangular matrices over a unital commutative ring R. A map φ on N is called preserving commutativity in both directions if xy = yx ? φ(x)φ(y) = φ(y)φ(x). In this paper, we prove that each invertible linear map on N preserving commutativity in both directions is exactly a quasi-automorphism of N, and a quasi-automorphism of N can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.     

Linear maps preserving regional eigenvalue location

Let M(n, C) be the vector space of n × n complex matrices and let G(r,s,t) be the set of all matrices in M(n, C) having r eigenvalues with positive real parts eigenvalues with negative real part and t eigenvalues with zero real part. In particularG(0,n,0) is the set of stable matrices. We investigate the set of linear operators on M(n, C) that map G(r,s,t) into itself. Such maps include, but are not always limited to similarities, transposition, and multiplication by a positive constant. The proof of our results depends on a characterization of nilpotent matrices in terms of matrices in a particular G(r,s,t), and an extension of a result about the existence of a matrix with prescribed eigenstructure and diagonal entries. Each of these results is of independent interest. Moreover, our char-acterization of nilpotent matrices is sufficiently general to allow us to determine the preservers of many other "inertia classes."     

Additive mappings preserving commutativity

The general form of additive surjective mappings on M_nwhich preserve commutativity in both directions is given.     

Linear approximation of approximately biorthogonality preserving maps

We investigate linear properties of mappings from a bounded domain of an n-dimensional normed space into another n-dimensional normed space such that the image of some almost biorthogonal system is almost biorthogonal. In this way we generalize a result of the author on stability of orthogonality in Euclidean spaces.     

Linear maps preserving the essential spectral radius

Let L(H) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert H. We characterize linear maps from L(H) onto itself that preserve the essential spectral radius.     

On strong commutativity preserving mappings

Let R be a ring, and S a non-empty subset of R. Suppose that R admits mappings F and G such that [F(x), G(y)] = [x,y] for all x, y ∈ S. In the present paper, we investigate commutativity of the ring R, when the mapping G is assumed to be a derivation or an endomorphism of R.     

Linear maps preserving the minimum and reduced minimum moduli

We describe linear maps from a C^∗-algebra onto another one preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the reduced minimum modulus.     

Linear maps preserving essential spectral functions and closeness of operator ranges

Let H be an infinite-dimensional complex Hilbert space and (H)the algebra of all bounded linear operators on H. Some characterizationsare obtained of linear maps on (H) preserving essential spectralfunctions such as the intersection of left essential spectrumand right essential spectrum of operators, preserving Fredholmoperators, and preserving semi-Fredholm operators, and preservingcloseness of operator ranges. A result of Mbekhta, Rodman andemrl (Integral Equations Operator Theory 55 (2006) 93-109) ongeneralized invertibility preserving linear maps is improved.     

Linear maps preserving Fredholm and Atkinson elements of C *-algebras

Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.     

Strong commutativity preserving generalized derivations on Lie ideals

We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f?:?L?→?R is a map and g is a generalized derivation of R such that [f(x),?g(y)]?=?[x,?y] for all x,?y?∈?L. Then there exist a nonzero α?∈?C and a map μ?:?L?→?C such that g(x)?=?αx for all x?∈?R and f(x)?=?α^?1 x?+?μ(x) for all x?∈?L, except when R???M ₂(F), the 2?×?2 matrix ring over a field F.     

Mappings preserving spectrum and commutativity on Hermitian matrices

The general form of a continuous mapping φ acting on the real vector space of all n × n complex Hermitian or real symmetric matrices, and preserving spectrum and commutativity, is derived. It turns out that φ is either linear or its image forms a commutative set.     

Strong commutativity preserving generalized derivations on right ideals

We characterize a prime ring R which admits a generalized derivation g and a map f : ρ → R such that [ f (x), g(y)]?=?[x, y] for all ${x,y\in \rho}$ , where ρ is a nonzero right ideal of R. With this, several known results can be either deduced or generalized.