Hyperbolic polynomials and spectral order

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