Upper Triangular Matrix Algebra Paul K-power Mapping

Let F be an arbitrary field of characteristic different from 2 and 3, and C be field of complex numbers. Suppose that T₂(F) and T₂(C) are the algebra of 2×2 upper triangular matrices on F and C respectively. If A∈T₂(F) satisfies A³ = A, then A is called a tripotent matrix. If A∈T₂(C) satisfies A^k = A, then A is called a k-potent matrix, where k≥3. We use T and T₂^k(C) to denote the subset of all tripotent matrices in T₂(F) and T₂(C) respectively. We will determine all injective mappingsφof T₂(F) which satisfies if A -λB∈T thenφ(A) -λφ(B)∈T, (?)λ∈F, and A, B∈T₂(F). Denote the set of all such mappingsφasΦ(F). A map is called to be preserving k-potent if it satisfies A-λB∈T₂^k(C), thenφ(A) -λφ(B)∈T₂^k(C), (?)λ∈C, and A, B∈T₂(C). We useΦ(C) to denote the set of all injective mapsφfrom T₂(C) to itself which preserving k-potent.In the chapter l,we give a brief introduction to additive,multiplicative,linear preserver problems and A -λB preserver problems. In the chapter 2, the forms of the injective maps from T₂(F) to T₂(F) preserving tripotent are characterized. In chapter 3, the forms of the injective maps from T₂(C) to T₂(C) preserving k-potent are characterized.Symbol E_ijstands for the matrix having the (i,j)-th entries equal to one and all other entries equal to zero, A^T stands for the transposition of A and I₂ stands for the identity matrix in T₂(F). Denote the set {1, 2,…, n} as [1,n] and A as {ε| s^k-1 = 1}. The mapφis bijective which is from [1,n] to [1,n]. (?) stands for the conjugate of A. A^* stands for (?) called conjugate transpose of A. R(X) stands for spectrum radiux of A. GL_n(F) is general linear group over F.