Linear equations in subspaces of operators

Given a subspace S of operators on a Hilbert space, and given two operators X and Y (not necessarily in S), when can we be certain that there is an op erator A in S such that AX = Y? If there is one, can we find some bound for its norm? These questions are the subject of a number of papers, some by t he present authors, and mostly restricted to the case where S is a reflexiv e algebra. In this paper, we relate the broader question involving operator subspaces to the question about reflexive algebras, and we examine a new m ethod of forming counterexamples, which simplifies certain constructions an d answers an unresolved question. In particular, there is a simple set of c onditions that are necessary for the existence of a solution in the reflexi ve algebra case; we show that-even in the case where the co-rank of X is on e-these conditions are not in general sufficient.