ИСТИНА

The talk is based on the joined work with Alexander Guterman and Artem Maksaev. The first result on linear preservers was obtained by Ferdinand Georg Frobenius, who characterized linear maps on complex matrix algebra preserving the determinant. Let Mn(F) be the n×n matrix algebra over a field F and Y be a subset of Mn(F). We say that a transformation T : Y → Mn(F) is of a standard form if there exist non-singular matrices P,Q such that T(A) = PAQ or T(A)=PATQ for all A∈Y. (1) Frobenius [1] proved that if T : Mn(C) → Mn(C) is linear and preserves the determinant, i.e., det(T(A)) = det(A) for all A ∈ Mn(C), then T is of the standard form (1) with det(PQ) = 1. In 1949 Jean Dieudonn´e [2] generalized this result for an arbitrary field F. He replaced the determinant preserving condition by the singularity preserving condition and proved the corresponding result for a bijective map T. In 2002 Gregor Dolinar and Peter ˇSemrl [3] modified the classical result of Frobenius by removing the linearity and replacing the determinant preserving condition by det(A+λB) = det(T(A)+λT(B)) for all A,B ∈ Mn(F) and all λ ∈ F (2) for F = C.Theyproved that if T : Mn(C) → Mn(C)is surjective and satisfies (2), then T is linear and hence is of the standard form (1) with det(PQ) = 1. Soon after that, Victor Tan and Fei Wang [4] generalized this proof for a field F with |F| > n and showed that under the condition (2) the map T is linear even without the surjectivity condition. Moreover, they revealed that if T is surjective, then only two different values of λ are required in (2). To be more precise, if |F| > n and T : Mn(F) → Mn(F) is a surjective map satisfying det(A +λiB) = det(T(A)+λiT(B)) for all A,B ∈ Mn(F) and i = 1,2, where λi= 0 and (λ1/λ2)k= 1 for 1 k n−2, then T is of the standard form (1). Nevertheless, this result was also further generalized by Constantin Costara [5]. Suppose |F| > n2 and λ0 ∈ F. Let T: Mn(F) → Mn(F) be a surjective map satisfying (2) only for one fixed value of λ = λ0 : det(A + λ0B) = det(T(A) + λ0T(B)) for all A,B ∈ Mn(F). Costara obtained that if λ0= −1, then such T is of the standard form (1) with det(PQ) = 1. For λ0 = −1, he showed that there exist P,Q ∈ GLn(F), det(PQ) = 1, and A0 ∈ Mn(F) such that T(A) = P(A+A0)Q or T(A)=P(A+A0)TQ for all A∈Y. The aim of this work is to relax the condition (2) for T. It has been revealed that if F is an algebraically closed field, then the conditions on determinant in the above results of Dieudonn´e or Tan and Wang can be replaced by less restrictive. The following result has been obtained. Theorem. Suppose Y = GLn(F) or Y = Mn(F), T: Y → Mn(F) is a map satisfying the following conditions: • for all A,B ∈ Y and λ ∈ F, the singularity of A + λB implies the singularity of T(A) + λT(B); • the image of T contains at least one non-singular matrix. Then T is of the standard form (1). (Note that in the theorem above det(PQ) possibly differs from 1.) References [1] G. Frobenius, ¨Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber. Deutsch. Akad. Wiss. (1897), pp. 994–1015. [2] D. J. Dieudonn´e, Sur une g´en´eralisation du groupe orthogonal ´a quatre variables, Arch. Math. 1 (1949), pp. 282–287. [3] G. Dolinar, P. ˇSemrl, Determinant preserving maps on matrix algebras, Linear Algebra Appl. 348 (2002), pp. 189–192. [4] V. Tan, F. Wang, On determinant preserver problems, Linear Algebra Appl., 369 (2003), pp. 311-317. [5] C. Costara, Nonlinear determinant preserving maps on matrix algebras, Linear Algebra Appl., 583 (2019), pp. 165–170.