Linear maps preserving certain unitarily invariant norms of tensor products of matrices

Abstract

Linear preserver problem is an active and popular research topic in matrix theory and functional analysis. The main goal of linear preserver problems is to characterise the structure of linear maps on matrix spaces or operator spaces that preserve certain functions, subsets or relations. Let Mn denote the n x n complex matrix space. The first linear preserver problem proposed by Frobenius in 1896 was to characterise linear maps Φ : Mn → Mn such that det(Φ(A)) = det(A) for all A ϵ Mn· In recent years, partly due to the development of quantum science, much attention has been paid to the study of linear maps leaving invariant tensor products or certain propositions of tensor products. Fosner et al. characterised linear preservers for Schatten p-norms and Ky Fan k- norms of tensor products of square matrices. In this thesis, we generalize their results by characterising the form of linear maps preserving the γ-norms or the (p, k)-norms with 2 < p < ∞ of tensor products of square matrices. Let m ≥ 2 and n1,..., nm be integers larger than or equal to 2. Suppose that ǁ·ǁ is the γ-norm or the (p, k)-norm with 2 < p < ∞. We show in this thesis that a linear map Φ : Mn1...nm → Mn1...nm satisfies ǁΦ(A1 x ··· x Am)ǁ = ǁA1 x ··· x Amǁ for all Ai ϵ Mni,i = 1,...,m, if and only if there exist unitary matrices U, V ϵ Mn1...nm such that Φ(A1 x...x Am) = U(φ1(A) x...x φm(A))V for all Ai ϵ Mni,i = 1,...,m, where φi is the identity map or the transposition map A → AT for i = 1,...,m. We develop some new techniques to show that Φ(Eii x Ejj) and Φ(Err x Ess) are orthogonal for any distinct (i, j) ≠ (r, s), which is a key step in our proof. Suppose that γ = (γ1,...,γn) with γ1 ≥ ... ≥ γk >0 = γk+1 = ··· = γn· Our characterization of linear preservers for γ-norms mainly relies on the observation that if ǁE + Fǁγ = ǁEǁγ + ǁFǁγ, then UEV = E1 + E2 and UFV = F1 + F2 for some unitary matrices U and V with E1, F1 ϵ Mk and E2, F2 ϵ Mn-k· Some equalities have been applied to obtain our results on (p, k)-norms.