Hyperbolic polynomials and spectral order

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