Approximation by Polynomials with Nonnegative Coefficients and the Spectral Theory of Positive Operators

For Σ a compact subset of C symmetric with respect to conjugation and f : Σ → C a continuous function, we obtain sharp conditions on f and Σ that insure that f can be approximated uniformly on Σ by polynomials with nonnegative coefficients. For X a real Banach space, K ⊆ X a closed but not necessarily normal cone with K −K = X, and A : X → X a bounded linear operator with A[K] ⊆ K, we use these approximation theorems to investigate when the spectral radius r(A) of A belongs to its spectrum σ(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 σ(i(A − A∗)) is zero, then r(A) ∈ σ(A). However, we also give an example of a normal operator A = −U − αI (where U is unitary and α > 0) for which A[K] ⊆ K and r(A) / ∈ σ(A).