Spectral order and isotonic differential operators of Laguerre-Polya type

The spectral order on $\bR^n$ induces a natural partial ordering on the manifold $\calH_{n}$ of monic hyperbolic polynomials of degree $n$. We show that all differential operators of Laguerre-P\'olya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space $\li$ of real bounded sequences. As a consequence, we deduce that the monoid $\calA'$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre-P\'olya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\calA'$-orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.