An operator equation on a Banach space, which represents the operator
analog of Burgers equation, is here considered. The well known Cole-Hopf
transformation, a particular case of the wider class of Backlund
transformations, which connects the classical nonlinear Burgers equation
to the linear heat equation, is extended to the case of operator valued
equations. Then, since the operator Burgers equation admits a recursion
operator, a whole hierarchy of Burgers operator equations is generated.
Notably, each member of such a Burgers operator hierarchy is related,
via Cole-Hopf transformation to the corresponding member of a heat
operator hierarchy. Indeed, also the recursion operator admitted by the
Burgers operator equation, is related, via Cole-Hopf transformation, to
the (trivial) recursion operator admitted by the linear heat operator
equation. Furthermore, the Burgers recursion operator is not Abelian,
hence, the whole hierarchy does not enjoy commutativity properties