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

For $\Sigma$ a compact subset of $\mathbf{C}$ symmetric with respect to conjugation and $f: \Sigma \to \mathbf{C}$ 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, $K \subseteq X$ a closed but not necessarily normal cone with $\overline{K - K} = X$, and $A: X \to X$ a bounded linear operator with $A[K] \subseteq K$, we use these approximation theorems to investigate when the spectral radius $\text{\rm r}(A)$ 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 $\text{\rm r}(A) \in \sigma (A)$. However, we also give an example of a normal operator $A = - U -\alpha I$ (where $U$ is unitary and $\alpha > 0$) for which $A[K] \subseteq K$ and $\text{\rm r}(A) \notin \sigma (A)$.