Linear equations in subspaces of operators

Abstract:

Given a subspace $\mathcal {S}$ of operators on a Hilbert space, and given two operators $X$ and $Y$ (not necessarily in $\mathcal {S}$), when can we be certain that there is an operator $A$ in $\mathcal {S}$ 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 the present authors, and mostly restricted to the case where $\mathcal {S}$ is a reflexive algebra. In this paper, we relate the broader question involving operator subspaces to the question about reflexive algebras, and we examine a new method of forming counterexamples, which simplifies certain constructions and answers an unresolved question. In particular, there is a simple set of conditions that are necessary for the existence of a solution in the reflexive algebra case; we show that — even in the case where the co-rank of $X$ is one — these conditions are not in general sufficient.