The solution set to the nonlinear operator equation XAX = BX on Hilbert Spaces 1

Abstract

We present the representation of solution sets of the operator equation XAX = BX on Hilbert spaces by taking advantage of the invariant  subspace of B, including solution sets of nonzero solutions, normal solutions and self-adjoint solutions. Moreover, necessary conditions  and sufficient conditions are established for existence of nonzero solutions of the operator equation XAX = BX which commute with B.