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

AbstractWe 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.Mathematics Subject Classification (2020): 47A62Key words: Operator equation*-orderinvariant subspacereducing subspace