Linear maps transforming the higher numerical ranges

Let k ∈{1...,n} The k-numerical range of A ∈ Mnis 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 ∈ Mnif 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→ Mnhas the form A→[(trA)In-(n-k)A]/k. Linear maps φ̃:Mn→Mn′ satisfying wk′(φ̃(A))=wk(A) for all A ∈ Mnare also studied. Furthermore, results are extended to triangular matrices.