Linear maps transforming the higher numerical ranges

Let k ∈ {1, . . . , n}. The k-numerical range of A ∈Mn is the set Wk(A) = {(trX∗AX)/k : X is n× k, X∗X = Ik}, and the k-numerical radius of A is the quantity wk(A) = max{|z| : z ∈ Wk(A)}. Suppose k > 1, k′ ∈ {1, . . . , n′} and n′ < C(n, k)min{k′, n′ − k′}. It is shown that there is a linear map φ : Mn → Mn′ satisfying Wk′(φ(A)) = Wk(A) for all A ∈ Mn if and only if n′/n = k′/k or n′/n = k′/(n−k) is a positive integer. Moreover, if such a linear map φ exists, then there is a unitary matrix U ∈Mn′ and nonnegative integers p, q with p+ q = n′/n such that φ has the form A 7→ U∗[A⊕ · · · ⊕ A } {{ } p ⊕A ⊕ · · · ⊕ A } {{ } q ]U or A 7→ U∗[ψ(A)⊕ · · · ⊕ ψ(A) } {{ } p ⊕ψ(A) ⊕ · · · ⊕ ψ(A) } {{ } q ]U, where ψ :Mn →Mn has the form A 7→ [(trA)In − (n− k)A]/k. Linear maps φ̃ :Mn →Mn′ satisfying wk′(φ̃(A)) = wk(A) for all A ∈ Mn are also studied. Furthermore, results are extended to triangular matrices. AMS Classifications: 15A04, 15A60, 47A12.