Linear maps relating different unitary similarity orbits or different generalized numerical ranges (Q1894509)

Let \(\mathcal M\) denote either the complex linear space \(M_n\) of \(n \times n\) complex matrices or the real linear space \(H_n\) of \(n \times n\) Hermitian matrices. The unitary similarity orbit of \(C \in {\mathcal M}\) is defined by \({\mathcal U} (C) = \{UCU^* : U \text{ unitary}\}\) and the corresponding circular unitary orbit is the set \({\mathcal V} (C) = \{\mu X : \mu\) a scalar, \(|\mu|= 1\), \(X \in {\mathcal U} (C)\}\). For \(A \in {\mathcal M}\) define the \(C\)-numerical range and the \(C\)-numerical radius by \(W_c (A) = \{\text{tr} (AX) : X \in {\mathcal U} (C)\}\) and \(r_c = \max \{|z|: z \in W_c(A)\},\) respectively. Let \(C, D \in H_n\). The authors study linear operators \(T\) on \(\mathcal M\) satisfying one of the following properties: (I) \(W_D(T(A)) = W_C (A)\) for all \(A \in {\mathcal M}\), (II) \(r_D (T(A)) = r_C(A)\) for all \(A \in {\mathcal M}\), (III) \({T({\mathcal U} (D)) = {\mathcal U} (C)}\), (IV) \(T({\mathcal V} (D)) = {\mathcal V} (C)\). In particular, they determine the conditions on \(C\) and \(D\) for the existence of a linear operator \(T\) satisfying any one of the conditions (I)-(IV), and characterize such an operator if it exists.