  Given a linear ordinary differential operator T with polynomial coefficients,
we study the class of closed subsets of the complex plane such that T sends any
polynomial (resp. any polynomial of degree exceeding a given positive integer)
with all roots in a given subset to a polynomial with all roots in the same
subset or to 0. Below we discuss some general properties of such invariant
subsets as well as the problem of existence of the minimal under inclusion
invariant subset.
