General preservers of quasi-commutativity on self-adjoint operators

Let H be a separable Hilbert space and Bsa(H) the set of all bounded linear self-adjoint operators. We say that A,B∈Bsa(H) quasi-commute if there exists a nonzero ξ∈C such that AB=ξBA. Bijective maps on Bsa(H) which preserve quasi-commutativity in both directions are classified.     

On projective equivalence of invariant subspace lattices

For i = 1,2, let A_i be a linear transformation on a complex vector space and let σ be a lattice isomorphism from the invariant subspace lattice of A₁ onto the invariant subspace lattice of A₂. We determine the conditions under which σ is implemented by a linear or conjugate linear transformation (or a sum of these two kinds).     

Elastic lattices: equilibrium, invariant laws and homogenization

In the recent years, lattice modelling proved to be a topic of renewed interest. Indeed, fields as distant as chemical modelling and biological tissue modelling use network models that appeal to similar equilibrium laws. In both cases, obtaining an equivalent continuous model allows to simplify numerical procedures. We define the basic properties of lattices: elasticity, frame-indifference, hyperelasticity. We determine rigorously the form that constitutive laws undertake under frame-indifference and hyperelasticity assumptions. Finally, we describe an homogenization technique designed for discrete structures that provides a limit continuum mechanics model and, in the special case of hexagonal lattices, we investigate the symmetry properties of the limit constitutive law.      

Linear preservers of isomorphic types of lattices of invariant operator ranges

We describe all linear self-mappings of the space of bounded linear operators in an infinite dimensional separable complex Hilbert space which preserve the isomorphism class of the lattice of invariant operator ranges.     

Three linear preservers on infinite triangular matrices

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.     

Rank preservers of matrices over max algebra

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We consider the semimodules over max algebra and study the properties of the weak basis and weak dimension of the semi-modules. Moreover, we obtain the characterizations of those linear operators that preserve rank of matrices over max-algebra.     

Rank preservers of matrices over max algebra

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We consider the semimodules over max algebra and study the properties of the weak basis and weak dimension of the semi-modules. Moreover, we obtain the characterizations of those linear operators that preserve rank of matrices over max-algebra.     

Noninvertibility preservers on Banach algebras

It is proved that a linear surjection Ф: A → B, which preserves noninvertibility between semisimple, unital, complex Banach algebras, is automatically injective.     

Additive preservers of Drazin invertible operators with bounded index

Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n ≥ 1, we show that an additive surjective map Φ on B(X)preserves Drazin invertible operators of index non-greater than n in both directions if and only if Φ is either of the form Φ(T) = αATA~(-1) or of the form Φ(T) = αBT~*B~(-1) where α is a non-zero scalar,A:X → X and B:X~*→ X are two bounded invertible linear or conjugate linear operators.     

Linear preservers and quantum information science

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 on mn?×?mn Hermitian matrices such that φ(A???B) and A???B have the same spectrum for any m?×?m Hermitian A and n?×?n Hermitian B. Such a map has the form A???B???U(?₁(A)????₂(B))U* for mn?×?mn Hermitian matrices in tensor form A???B, where U is a unitary matrix, and for j?∈?{1,?2}, ? _j is the identity map?X???X or the transposition map?X???X ^t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A???B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.     

Minus partial order and linear preservers

In this paper, we investigate the properties of minus partial order in unital rings. We generalize several results well known for matrices and bounded linear operators on Banach spaces. We also study linear maps preserving the minus partial order in unital semisimple Banach algebras with essential socle.     

Modular lattices over cyclotomic fields

This paper is concerned with modular lattices over cyclotomic fields. In particular, the notion of Arakelov modular ideal lattice is introduced. All the cyclotomic fields over which there exists an Arakelov modular lattice of given level are characterised.     

Bounds for solid angles of lattices of rank three

We find sharp absolute constants C₁ and C₂ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C₁,C₂]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in RN with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R³. Such spherical configurations come up in connection with the kissing number problem.     

Linear maps leaving the alternating group invariant

Let A_n be the group of n×n even permutation matrices, and let V_n be the real linear space spanned by A_n. The purpose of this note is to characterize those linear operators φ on V_n satisfying φ(A_n)=A_n. This answers a question raised by C.K. Li, B.S. Tam, N.K. Tsing [Linear Algebra Appl., to appear].