Involution-preserving maps without the linearity assumption and its application

다국어 초록 (Multilingual Abstract)

Suppose F is a field of characteristic not 2 and F ≠ Z3. Let
Mn(F) be the linear space of all n×n matrices over F, and let Γn(F) be the
subset of Mn(F) consisting of all n×n involutory matrices. We denote by
Φn(F) the set of all maps from Mn(F) to itself satisfying A−λB∈ Γn(F)
if and only if ∅(A)− λ∅(B)∈ Γn(F) for every A,B ∈ Mn(F) and λ ∈ F. It
was showed that ∅∈Φn(F) if and only if there exist an invertible matrix
P ∈ Mn(F) and an involutory element ε such that either ∅(A) = εPAP−1
for every A ∈ Mn(F) or ∅(A) = εPAT P−1 for every A ∈ Mn(F). As an
application, the maps preserving inverses of matrces also are characterized.

다국어 초록 (Multilingual Abstract)

Suppose F is a field of characteristic not 2 and F ≠ Z3. Let
Mn(F) be the linear space of all n×n matrices over F, and let Γn(F) be the
subset of Mn(F) consisting of all n×n involutory matrices. We denote by
Φn(F) the set of all maps from Mn(F) to itself satisfying A−λB∈ Γn(F)
if and only if ∅(A)− λ∅(B)∈ Γn(F) for every A,B ∈ Mn(F) and λ ∈ F. It
was showed that ∅∈Φn(F) if and only if there exist an invertible matrix
P ∈ Mn(F) and an involutory element ε such that either ∅(A) = εPAP−1
for every A ∈ Mn(F) or ∅(A) = εPAT P−1 for every A ∈ Mn(F). As an
application, the maps preserving inverses of matrces also are characterized.

참고문헌 (Reference)

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