Gregor Dolinar

Determinant preserving maps on matrix algebras

Abstract Let M n be the algebra of all n × n complex matrices. If φ : M n → M n is a surjective mapping satisfying det( A + λB )=det( φ ( A )+ λφ ( B )), A , B ∈ M n , λ∈ C , then either φ is of the form φ ( A )= MAN , A ∈ M n , or φ is of the form φ ( A )= MA t N , A ∈ M n , where M , N ∈ M n are nonsingular matrices with det( MN )=1.

Role of phospholipid asymmetry in the stability of inverted hexagonal mesoscopic phases.

The role of phospholipid asymmetry in the transition from the lamellar (L(alpha)) to the inverted hexagonal (H(II)) phase upon the temperature increase was considered. The equilibrium configuration of the system was determined by the minimum of the free energy including the contribution of the isotropic and deviatoric bending and the interstitial energy of phospholipid monolayers. The shape and local interactions of a single lipid molecule were taken into account. The minimization with respect to the configuration of the lipid layers was performed by a numerical solution of the system of the Euler-Lagrange differential equations and by the Monte Carlo simulated annealing method. At high enough temperature, the lipid molecules attain a shape exhibiting higher intrinsic mean and deviatoric curvatures, which fits better into the H(II) phase than into the L(alpha) phase. Furthermore, the orientational ordering of lipid molecules in the curvature field expressed as the deviatoric bending provides a considerable negative contribution to the free energy, which stabilizes the nonlamellar H(II) phase. The nucleation configuration for the L(alpha)-H(II) phase transition is tuned by the isotropic and deviatoric bending energies and the interstitial energy.

Maps on matrix algebras preserving idempotents

Abstract Let M n be the algebra of all n × n complex matrices and P n the set of all idempotents in M n . Suppose φ : M n → M n is a surjective map satisfying A − λB ∈ P n if and only if φ ( A )− λφ ( B )∈ P n , A , B ∈ M n , λ∈ C . Then either φ is of the form φ ( A )= TAT −1 , A ∈ M n , or φ is of the form φ ( A )= TA t T −1 , A ∈ M n , where T ∈ M n is a nonsingular matrix.

Maps on upper triangular matrices preserving Lie products

Let be an arbitrary field with characteristic zero, let Tn be the Lie algebra of all n × n upper triangular matrices over with the Lie product [A,B] = A B − B A, and let a bijective map φ : Tn → Tn satisfy φ ([A,B]) = [φ (A) , φ (B)], A,B ∈ Tn . Then there exist an invertible matrix T ∈ Tn , a function satisfying ϕ (C) =0 for every strictly upper triangular matrix C ∈ Tn , and an automorphism f of the field , such that for all [aij ]∈ Tn , or for all [aij ] ∈ Tn , where .

Maps on Matrix Algebras Preserving Commutativity

Let n≥3 and let Mn be the algebra of all n×n complex matrices. If is a surjective mapping satisfying if and only if , then there exist a nonzero scalar c, an invertible matrix T, a function , and an automorphism f of the field , such that either , or .

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.

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.

Maps on quantum observables preserving the gudder order

Let B s ( H ) be the space of all self-adjoint bounded linear operators on a complex Hilbert space H . The elements of B s ( H ) represent bounded quantum observables. Recently, Gudder [3] introduced a new order ≤ G on B s ( H ). In this paper we determine the general form of all continuous automorphisms of the poset ( B s ( H ), ≤ G ).

Maps on B(H) Preserving Idempotents

Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H and the set of all idempotents in B(H). Suppose is a surjective map satisfying if and only if . Then ϕ is either of the form and T is a continuous invertible linear operator on H, or of the form and T is a continuous invertible conjugate linear operator on H.