# Spectral functions with sign-preserving properties

Let $$\mathcal{S}_n$$ denote the space of symmetric positive definite matrices. Suppose $$\mathbf{R} \in \mathcal{S}_n$$ has smallest eigenvalue $$\lambda_{min} > 1/2$$ and is entry-wise positive. Given some scalar function $$f$$, define the corresponding spectral function $$F(\mathbf{R}):\mathcal{S}_n \to \mathcal{S}_n$$ such that $$F(\mathbf{R})$$ shares its eigenvectors with $$\mathbf{R}$$ yet the eigenvalues are replaced by $$f(\lambda_i)$$, respectively. Fix some $$n$$-dimensional non-negative vector $$\mathbf{x}$$.

For which spectral functions $$F$$ do we have $$F(\mathbf{R}) \mathbf{x} \geq 0$$?

It is known that non-negativity of matrices of general dimension is preserved iff $$f$$ is absolutely monotonic. While sufficient, I wonder whether a weaker condition will do. Something that may matter is that the angle between a vector and its image under a positive-definite matrix is bounded by 90°.

I further wonder whether additional assumptions on $$\mathbf{R}$$ may help, e.g. strict diagonal dominance.