Linear maps transforming the higher numerical ranges

Let k ∈ {1, … , n}. The k-numerical range of A ∈ Mn is the setWk(A)={(trX*AX)/k:Xisn×k,X*X=Ik},and the k-numerical radius of A is the quantitywk(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 are unitary matrix U∈Mn′ and nonnegative integers p, q with p + q = n′/n such that ϕ has the formA↦U*[A⊕⋯⊕A︸p⊕At⊕⋯⊕At︸q]UorA↦U*[ψ(A)⊕⋯⊕ψ(A)︸p⊕ψ(A)t⊕⋯⊕ψ(A)t︸q]U,where ψ : Mn → Mn has the form A↦[(trA)In-(n-k)A]/k. Linear maps ϕ˜:Mn→Mn′ satisfying wk′(ϕ˜(A))=wk(A) for all A ∈ Mn are also studied. Furthermore, results are extended to triangular matrices.