Journal of the
Korean Mathematical Society JKMS

In this paper we obtained the following : Let $\mathcal H$ be a Hilbert space and $\mathcal L$ be a subspace lattice on $\mathcal H$. Let $X$ and $Y$ be operators acting on ${\mathcal H}$. If the range of $X$ is dense in $\mathcal H$, then the following are equivalent:
(1) there exists an operator $A$ in Alg$\mathcal L$ such that $AX=Y$,
(2) $\displaystyle\sup\left\{ {{\|E^\bot Yf\|} \over {\|E^\bot Xf\|}} : f\in {\mathcal H}, ~E\in {\mathcal L} \right \}=K<\infty.$
Moreover, if condition (2) holds, we may choose the operator $A$ such that $\|A\|=K$.

Keywords: interpolation problem, subspace lattice, Alg$\mathcal L$, CSL

MSC numbers: 47L35