Linear maps transforming the higher numerical ranges

Abstract

Let k ∈ {1, … , n}. The k-numerical range of A ∈ M n is the set W k ( A ) = { ( tr X * AX ) / k : X is n × k , X * X = I k } , and the k-numerical radius of A is the quantity w k ( A ) = max { | z | : z ∈ W k ( 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 : M n → M n ′ satisfying W k ′ ( ϕ ( A ) ) = W k ( A ) for all A ∈ M n 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 ∈ M n ′ and nonnegative integers p, q with p + q = n′/ n such that ϕ has the form A ↦ U * [ A ⊕ ⋯ ⊕ A ︸ p ⊕ A t ⊕ ⋯ ⊕ A t ︸ q ] U or A ↦ U * [ ψ ( A ) ⊕ ⋯ ⊕ ψ ( A ) ︸ p ⊕ ψ ( A ) t ⊕ ⋯ ⊕ ψ ( A ) t ︸ q ] U , where ψ : M n → M n has the form A ↦ [ ( tr A ) I n - ( n - k ) A ] / k . Linear maps ϕ ˜ : M n → M n ′ satisfying w k ′ ( ϕ ˜ ( A ) ) = w k ( A ) for all A ∈ M n are also studied. Furthermore, results are extended to triangular matrices.