Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

For Sigma a compact subset of C symmetric with respect to conjugation and f : Sigma --> C a continuous function, we obtain sharp conditions on f and Sigma that insure that f can be approximated uniformly on Sigma by polynomials with nonnegative coefficients. For X a real Banach space,K subset of or equal to X a closed but not necessarily normal cone with <(K - K)over bar> = X, and A : X --> X a bounded linear operator with A[K] subset of or equal to K, we use these approximation theorems to investigate when the spectral radius r(A) of A belongs to its spectrum sigma(A). A special case of our results is that if X is a Hilbert space, A is normal and the 1-dimensional Lebesgue measure of sigma(i(A - A*)) is zero, then r(A) epsilon sigma(A). However, we also give an example of a normal operator A = - U - alpha I (where U is unitary and alpha > 0) for which A[K] subset of or equal to K and r(A) is not an element of sigma(A).