Linear maps on matrices: the invariance of symmetric polynomials

Let L be a linear map on the space M_n of all n by n complex matrices. Let h(x₁,…,x_n) be a symmetric polynomial. If X is a matrix in M_n with eigenvalues λ₁,…,λ_n, denote h(λ₁,…,λ_n) by h(X). For a large class of polynomials h, we determine the structure of the linear maps L for which h(X)=h(L(X)), for all X in M_n.