Sufficient conditions for a linear operator on $\mathbb{R}[x]$ to be monotone

TL;DR

This paper investigates the relationship between hyperbolicity preservation and monotonicity in infinite order differential operators on real polynomials, providing counterexamples and sufficient conditions for monotonicity.

Contribution

It disproves the conjecture that hyperbolicity preserver implies monotonicity and offers new criteria for monotonicity in these operators.

Findings

Hyperbolicity preservation does not imply monotonicity.

Counterexamples to the conjecture are provided.

Sufficient conditions for monotonicity are established.

Abstract

We demonstrate that being a hyperbolicity preserver does not imply monotonicity for infinite order differential operators on $\mathbb{R}[x]$, thereby settling a recent conjecture in the negative. We also give some sufficient conditions for such operators to be monotone.