Bojan Kuzma

Additive mappings decreasing rank one

Abstract Let B ( X ) be the algebra of bounded operators on a real or complex Banach space X , and F( X ) a subalgebra of finite rank operators. A complete description of additive mappings Φ:F( X )→F( X ) , which map rank one operators to operators of rank at most one, is given.

On maximal distances in a commuting graph

We study maximal distances in the commuting graphs of matrix algebras defined over algebraically closed fields. In particular, we show that the maximal distance can be attained only between two nonderogatory matrices. We also describe rank-one and semisimple matrices using the distances in the commuting graph.

Maps preserving matrix pairs with zero Jordan product

We study the maps on complex Hermitian and complex symmetric matrices which preserve zeros of a Jordan product.

Commuting graphs and extremal centralizers

We determine the conditions for matrix centralizers which can guarantee the connectedness of the commuting graph for the full matrix algebra M n ( F ) over an arbitrary field F . It is known that if F is an algebraically closed field and n  ≥ 3 , then the diameter of the commuting graph of M n ( F ) is always equal to four. We construct a concrete example showing that if F is not algebraically closed, then the commuting graph of M n ( F ) can be connected with the diameter at least five.

On the Polya permanent problem over finite fields

Let F be a finite field of characteristic different from 2. We show that no bijective map transforms the permanent into the determinant when the cardinality of F is sufficiently large. We also give an example of a non-bijective map when F is arbitrary and an example of a bijective map when F is infinite which do transform the permanent into the determinant. The technique developed allows us to estimate the probability of the permanent and the determinant of matrices over finite fields having a given value. Our results are also true over finite rings without zero divisors.

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.

Preserving Zeros of a Polynomial

We study nonlinear surjective mappings on ℳ n (𝔽) and its subsets, which preserve the zeros of some fixed polynomials in noncommuting variables.

On diameter of the commuting graph of a full matrix algebra over a finite field

It is shown that the commuting graph of a matrix algebra over a finite field has diameter at most five if the size of the matrices is not a prime nor a square of a prime. It is further shown that the commuting graph of even-sized matrices over finite field has diameter exactly four. This partially proves a conjecture stated by Akbari, Mohammadian, Radjavi, and Raja Linear Algebra Appl. 418 (2006) 161-176.

A note on partial orders of Hartwig, Mitsch, and Šemrl

We show that on Rickart rings the partial orders of Mitsch and Semrl are equivalent. In particular, these orders are equivalent on B(H), the algebra of all bounded linear operators on a Hilbert space H.

Additive mappings that do not increase minimal rank of alternate matrices

Additive mappings, which do not increase the minimal rank of alternate matrices, are completely classified. No condition is imposed on the underlying field.Additive mappings, which do not increase the minimal rank of alternate matrices, are completely classified. No condition is imposed on the underlying field.