Linear maps relating different unitary similarity orbits or different generalized numerical ranges

Abstract

Let M be the complex linear space Mn of n × n complex matrices or the real linear space Hn of n × n hermitian matrices. For C ∈ M, its unitary similarity orbit is the set. U(C) = {UCU*; U unitary. and its circular unitary similarity orbit is the set. V(C) = {μX : μ ∈ F, |μ| = 1, X ∈ U(C)}. where F is the scalar field C or R according as M = Mn or M = Hn. Related to U(C) and V(C) are the C-numerical range and the C-numerical radius of A ∈ M defined by. Wc(A) = {tr(AX) : X ∈U(C)}. and. rC(A)=max{|z|:z∈WC(A)},. respectively. Let C, D ∈ Hn, we study the linear operators T on M satisfying one of the following properties: (I) WD(T(A)) = WC(A) for all A ∈ M, (II) rD(T(A)) = rC(A) for all A ∈ M, (III) T(U(D)) = U(C), (IV) T(V(D)) = V(C). In particular, we determine the conditions on C and D for the existence of a linear operator T on M satisfying any one of the conditions (I)-(IV), and characterize such an operator if it exists. © 1995.