Undergraduate Research Opportunity Programme in Science SOME PROBLEMS ON LINEAR PRESERVERS

The classification of preservers began about 100 years ago. In 1897, Frobenius characterized the linear operators on Mn which preserve certain matrix functions: those linear operators on Mn preserves the determinant and those preserves the characteristic polynomial (spectrum). He proved that if M M n n → : T is a linear transformation satisfying) det()) (T det(A A = , M A n ∈ , then either UAV A =) (T , M A n ∈ or V UA A T) (T = , M A n ∈ for some M V U n ∈ , with 1 det(= UV). Actually, linearity is a very strong property. In 2002, Dolinar and Semrl improved the classical result that surjective transformation M M n n → : T satisfies) det()) (T) (T det(B A B A λ λ + = + for all M B A n ∈ , , C ∈ λ have the same form. In Chapter 2, I remove the " surjective " assumption, and successfully prove that T is linear. It immediately follows that T is of the same form as well. (Theorem 2.3) It is already been clarified the linear transformations defined on Un , the space of n n × upper triangular matrices. As what I did for the determinate preserver, I replace the " linearity " by) det()) (T) (T det(B A B A λ λ + = + for all Un ∈ B A, , C ∈ λ , and successfully prove that there exist scalars c c n , , 1 L with 1 1 = ∏ = n i i c and permutation σ of } , , 1 { n L such that) () () ()] (T [ i i i ii c σ σ A A = for all n i , , 1 L = for all Un ∈ A. (Theorem 2.8) In 1986, Johnson and Pierce proved the general form of the linear rank-1 preservers on H n , the space of n n × hermitian matrices. That is, if T is a linear rank-1 preserver on H n with 2) T (≥ rank , then there exists an invertible matrix S and } 1 , 1 { − ∈ ε such that either H) (T SAS A ε = for all H n ∈ A or H T) (T S SA A ε = for all H …