Search

Let $$ \mathbb{F} $$ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and for the whole matrix space M( $$ \mathbb{F} $$). It is known that for n = 2, there are bijective linear maps Φ on $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and M( $$ \mathbb{F} $$) satisfying the condition per A = det Φ( A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ , ϕ), where Φ is an arbitrary bijective map on matrices and ϕ : $$ \mathbb{F} $$ → $$ \mathbb{F} $$ is an arbitrary map such that per A = ϕ(det Φ( A)) for all matrices A from the spaces $$ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $$ and M( $$ \mathbb{F} $$), respectively. Moreover, for the space M( $$ \mathbb{F} $$), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $$ \mathbb{F} $$ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples. [ABSTRACT FROM AUTHOR]

We study the maps on complex Hermitian and complex symmetric matrices which preserve zeros of a Jordan product. [ABSTRACT FROM AUTHOR]

Abstract: Let be a field and . Suppose contain all rank-one idempotents. The structure of surjections satisfying is determined. Similar results are also obtained for (a) subsets of bounded operators acting on a complex or real Banach space , (b) the space of Hermitian matrices acting on n-dimensional vectors over a skew-field , (c) subsets of self-adjoint bounded linear operators acting on an infinite dimensional complex Hilbert space. It is then illustrated that the results can be applied to characterize mappings ϕ on matrices or operators such thatfor functions F such as the spectral norm, Schatten p-norm, numerical radius and numerical range, etc. [Copyright &y& Elsevier]

Abstract: An operator T on the separable infinite-dimensional Hilbert space is constructed so that the commutant of every operator which is not a scalar multiple of the identity operator and commutes with T coincides with the commutant of T. On the other hand, it is shown that for several classes of operators it is possible to construct a finite sequence of operators, starting at a given operator from the class and ending in a rank-one projection such that each operator in the sequence commutes with its predecessor. The classes which we study are: finite-rank operators, normal operators, partial isometries, and contractions. It is also shown that for any given set of yes/no conditions between points in some finite set, there always exist operators on a finite-dimensional Hilbert space such that their commutativity relations exactly satisfy those conditions. [Copyright &y& Elsevier]