  In this paper we give another characterization of the strictly nilpotent
elements in the Weyl algebra, which (apart from the polynomials) turn out to be
all bispectral operators with polynomial coefficients. This also allows to
reformulate in terms of bispectral operators the famous conjecture, that all
the endomorphisms of the Weyl algebra are automorphisms (Dixmier, Kirillov,
etc).
