Operator Solutions of the Multi–valued Operator Equation $$A = BXC$$

Abstract Assume that A , B and C are three linear relations on certain linear spaces. The main objective of this note is to present some necessary and sufficient conditions for the existence of the solutions X of the operator equation $$A = BXC$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> . The obtained characterizations extend and complete some known operator factorization results due to R.G. Douglas, Z. Sebestyén and D. Popovici.