Tatjana Petek

Adjacency preserving maps on matrices and operators

We characterize injective continuous maps on the space of real or complex rectangular matrices preserving adjacent pairs of matrices. We also extend Huas fundamental theorem of the geometry of rectangular matrices to the infinite-dimensional case. An application in the theory of local automorphisms is presented.

Characterization of jordan homomorphisms on Mn using preserving properties

Abstract Let o: Mn → Mn, n ⩾ 3, be a continuous mapping preserving spectrum and commutativity in both directions (neither linearity nor multiplicativity of ☎i; is assumed). Then ☎i; is either an automorphism or an antiautomorphism. The same result holds if ☎i; preserves spectrum, commutativity, and rank one matrices.

Additive mappings preserving commutativity

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

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.

Norm preservers of Jordan products

Norm preserver maps of Jordan product on the algebra Mn of n×n complex matrices are studied, with respect to various norms. A description of such surjective maps with respect to the Frobenius norm is obtained: Up to a suitable scaling and unitary similarity, they are given by one of the four standard maps (identity, transposition, complex conjugation, and conjugate transposition) on Mn, except for a set of normal matrices; on the exceptional set they are given by another standard map. For many other norms, it is proved that, after a suitable reduction, norm preserver maps of Jordan product transform every normal matrix to its scalar multiple, or to a scalar multiple of its conjugate transpose.

Mappings preserving the idempotency of products of operators

We obtain a general form of a surjective (not assumed additive) mapping φ, preserving the nonzero idempotency of a certain product, being defined (a) on the algebra of all bounded linear operators B(X), where X is at least three-dimensional real or complex Banach space, (b) on the set of all rank-one idempotents in B(X) and (c) on the set of all idempotents in B(X). In any of the cases it turns out that φ is additive and either multiplicative or antimultiplicative.

Spectrum and commutativity preserving mappings on triangular matrices

We give the description of continuous mappings on the algebra of all upper-triangular complex matrices that preserve spectrum and commutativity.

Conditions for Linear Dependence of Two Operators

The linear dependence property of two Hilbert space operators is expressed in terms of equality of size of values of certain sesquilinear and quadratic forms associated with the operators. The forms are based on qnumerical ranges.

Reversibility in polynomial systems of ODE’s

Abstract For a given family of real planar polynomial systems of ordinary differential equations depending on parameters, we consider the problem of how to find the systems in the family which become time-reversible after some affine transformation. We first propose a general computational approach to solve this problem, and then demonstrate its usage for the case of the family of quadratic systems.

Spectrum and commutativity preserving mappings on H2

Abstract The continuous mappings on 2×2 real symmetric or complex hermitian matrices that are spectrum and commutativity preserving are characterized.