Interpolation in inflated Hilbert spaces

The interpolation problem for a reflexive algebra Alg$\mathcal{L}$ is this: Given two operators $X$ and $Y$, under what conditions can we be sure that there will exist an operator $A$ in Alg$\mathcal{L}$ such that $AX=Y$? There are simple necessary conditions that have been investigated in several earlier papers. Here we present an example to show that the conditions are not, in general, sufficient. We also suggest a strengthened set of conditions which are necessary and are ``almost'' sufficient, in the sense that they will ensure that $Y$ lies in the weak-operator closure of the set {$AX$:$A \in$Alg$\mathcal{L}$}.