Hyperbolic polynomials and spectral order

The spectral order on $\text{bR}$ induces a partial ordering on the manifold ${\text{calH}}_n$ of monic hyperbolic polynomials of degree $n$. We show that the semigroup $\widetilde{\text{calS}}$ generated by differential operators of the form $(1-\text{la}\frac{d}{dx})e^{\text{la}\frac{d}{dx}}$, $\text{la}\in \text{bR}$, acts on the poset ${\text{calH}}_n$ in an order-preserving fashion. We also show that polynomials in ${\text{calH}}_n$ are global minima of their respective $\widetilde{\text{calS}}$-orbits and we conjecture that a similar result holds even for complex polynomials. Finally, we show that only those pencils of polynomials in ${\text{calH}}_n$ which are of logarithmic derivative type satisfy a certain local minimum property for the spectral order.