Quadratic Hyperbolicity Preservers & Multiplier Sequences

It is known (see \cite[Brändén, Lemma 2.7]{B10}) that a necessary condition for $T:=\sum Q_k(x) D^k$ to be hyperbolicity preserving is that $Q_k(x)$ and $Q_{k-1}(x)$ have interlacing zeros. We characterize all quadratic linear operators, as a consequence we find several classes of $P_n$-multiplier sequence.

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