Characterize linear operators preserving zeros in the unit interval

Determine the semigroup of all linear operators T: C[z] -> C[z] that send every polynomial whose zeros lie in the unit interval on the real line (for example, [0,1]) to a polynomial whose zeros also lie in the unit interval (or to 0).

Background

Problem 1 in the paper asks, for a subset S ⊂ C, to describe all linear operators T:C[z]→C[z] that map polynomials with zeros in S to polynomials with zeros in S (or to 0). This is the classical Pólya–Schur-type classification problem.

The paper notes that this problem has been solved for circular domains (images of the unit disk under Möbius transformations), their boundaries, and more recently for strips, but not for the unit interval. The authors highlight that even this closely related real case remains unresolved.