The star-shapedness of a generalized numerical range

Let HnHn be the set of all n×nn×n Hermitian matrices and Hnm be the set of all m  -tuples of n×nn×n Hermitian matrices. For A=(A1,...,Am)∈Hnm and for any linear map L:Hnm→Rℓ, we define the L-numerical range of A byWL(A):={L(U⁎A1U,...,U⁎AmU):U∈Cn×n,U⁎U=In}.WL(A):={L(U⁎A1U,...,U⁎AmU):U∈Cn×n,U⁎U=In}. In this paper, we prove that if ℓ≤3ℓ≤3, n≥ℓn≥ℓ and A1,...,AmA1,...,Am are simultaneously unitarily diagonalizable, then WL(A)WL(A) is star-shaped with star center at L(trA1nIn,...,trAmnIn).