Solvability of certain quadratic operator equations and representations of Drazin inverses

Let E be a Banach space, B(E) be the set of bounded linear operators on E, and I be the identity operator on E. In this paper the solvability of the quadratic operator equations AX+X2=B and XA+I=XBX is studied, where A∈B(E) is invertible and B∈B(E) is nilpotent. It is proved that the first quadratic operator equation is solvable if furthermore AB=BA or E=Cn for n∈{2,3} and B2=0. A counterexample is given such that the second quadratic operator equation is unsolvable. As an application, in the general setting of bounded linear operators on Banach spaces, a new expression of the Drazin inverse is given under the condition that A,B∈B(E) are Drazin invertible and AB=BA.