Linear maps transforming the higher numerical ranges Original Research Article

Let k set membership, variant {1, … , n}. The k-numerical range of A set membership, variant Mn is the setimageand the k-numerical radius of A is the quantitywk(A)=max{z:zset membership, variantWk(A)}.Suppose k > 1, k′ set membership, variant {1, … , n′} and n′ < C(n, k)min{k′, n′ − k′}. It is shown that there is a linear map phi:Mn→Mn′ satisfying Wk′(phi(A))=Wk(A) for all A set membership, variant 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 phi exists, then there are unitary matrix Uset membership, variantMn′ and nonnegative integers p, q with p + q = n′/n such that phi has the formimageorimagewhere ψ : Mn → Mn has the form image. Linear maps image satisfying image for all A set membership, variant Mn are also studied. Furthermore, results are extended to triangular matrices.