Page 214 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 214
RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
for F = C. They proved 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(P Q) = 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(P Q) = 1.
For λ0 = −1, he showed that there exist P, Q ∈ GLn(F), det(P Q) = 1, and A0 ∈ Mn(F)
such that
T (A) = P (A + A0)Q or T (A) = P (A + A0)T Q 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 Dieudonné
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(P Q) possibly differs from 1.)
References
[1] G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen,
Sitzungsber. Deutsch. Akad. Wiss. (1897), pp. 994–1015.
[2] D. J. Dieudonné, Sur une généralisation du groupe orthogonal á quatre variables, Arch.
Math. 1 (1949), pp. 282–287.
[3] G. Dolinar, P. Šemrl, 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.
212
for F = C. They proved 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(P Q) = 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(P Q) = 1.
For λ0 = −1, he showed that there exist P, Q ∈ GLn(F), det(P Q) = 1, and A0 ∈ Mn(F)
such that
T (A) = P (A + A0)Q or T (A) = P (A + A0)T Q 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 Dieudonné
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(P Q) possibly differs from 1.)
References
[1] G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen,
Sitzungsber. Deutsch. Akad. Wiss. (1897), pp. 994–1015.
[2] D. J. Dieudonné, Sur une généralisation du groupe orthogonal á quatre variables, Arch.
Math. 1 (1949), pp. 282–287.
[3] G. Dolinar, P. Šemrl, 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.
212
