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Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of $\mathcal{D} h$?

Some examples:

$\bullet$ $\mathcal{D} h = f(x) h(x)$, where $f$ is a polynomial all of whose roots are in $[0,1]$.

$\bullet$ $\mathcal{D} = D_x$, and more generally $D_x^k$, by Rolle's theorem.

$\bullet$ $\mathcal{D} h = (x-r)^{1-a} D_x \left( (x-r)^a h \right)$ for $r \in [0,1]$ and $0 \leq a$. By Rolle's theorem applied to $(x-r)^a h$. Properly, this only makes sense when $a$ is of the form $m/(2n+1)$, so that we can raise negative numbers to the $a$, but formally it gives a differential operator for all $a$ and it preserves real-rootedness by continuity.

$\bullet$ $\mathcal{D} h = (x-r)^{k-a} D_x^k \left( (x-r)^a h \right)$ for $r \in [0,1]$ and $k-1 < a$, generalizing the above.

$\bullet$ $\mathcal{D} h = \left( \prod_{i=1}^s (x-r_i)^{k-a_i} \right) \cdot D_x^k \left( \prod_{i=1}^s (x-r_i)^{\vphantom{k}a_i} h \right)$ for $0 \leq r_1 \leq r_2 \leq \cdots \leq r_s \leq 1$ and $a_1$, $a_2$, ..., $a_s \geq k-1$, generalizing the above, if I didn't make any errors.

$\bullet$ Any composition of the above.

Motivation: I just wrote an answer which came down to manipulating a differential operator until I could show that it had this property. (The question was about polynomials in $x^{-1}$ all of whose roots are in $[2, \infty)$, but that's easily equivalent.) I'm curious whether there is an algorithm for this sort of thing.

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Perhaps you know this, but the if you had demanded instead that $\mathcal D$ preserves the property of having all roots in $\mathbb R$, then the result would be one of the main theorems in Borcea and Brändén, Multivariate Pólya-Schur classification problems in the Weyl algebra, Proc. Lond. Math. Soc. (3) 101 (2010). –  Dan Petersen Jul 17 '12 at 15:50
    
Nope, didn't know that. Thanks! I wouldn't be surprised if there were some clever trick to turn one into the other. –  David Speyer Jul 17 '12 at 15:53
    
Also look at their Annals paper annals.math.princeton.edu/2009/170-1/p14 Their main result characterizes linear operators which preserve real rooted polynomials as the ones whose symbol is a real stable polynomial. The latter can be further characterized as a determinant by another theorem of the authors. –  Gjergji Zaimi Jul 17 '12 at 17:07
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Notice that the corresponding classification problem for positive reals is open. –  Gjergji Zaimi Jul 17 '12 at 17:22
    
Thanks! The authors state (last paragraph, section 4) that an answer to the question about closed intervals would "have many interesting applications". Sounds like this is open for the time being. –  David Speyer Jul 17 '12 at 17:49
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