Journal of the
Korean Mathematical Society JKMS

Let $\mathcal L$ be a subspace lattice on a Hilbert space $\mathcal H$ and $X$ and $Y$ be operators acting on a Hilbert space
${\mathcal H}$. Let $P$ be the projection onto $\overline{{\mathcal R} (X)}$, where ${\mathcal R} X$ is the range of $X$. If $PE=EP$ for each $E\in {\mathcal L}$, then there exists an operator $A$ in Alg$\mathcal L$ such that $AX=Y$ if and only if
$$\displaystyle\sup\{ {{\|E^\bot Yf\|} / {\|E^\bot Xf\|}} : f\in {\mathcal H}, ~E\in {\mathcal L} \}= K <\infty.$$ Moreover, if the necessary condition holds, then we may choose an operator $A$ such that $AX=Y$ and $\|A\| = K$.
Let $x$ and $y$ be vectors in ${\mathcal H}$ and let $P_x$ be the projection onto the singlely generated space by $x$. If $P_xE=EP_x$ for each $E\in {\mathcal L}$, then the assertion that there exists an operator $A$ in Alg$\mathcal L$ such that $Ax=y$ is equivalent to the condition $$K_0 : = \displaystyle\sup\{{{\|E^\bot y\|} / {\|E^\bot x\|}} : ~E\in {\mathcal L} \} <\infty.$$ Moreover, we may choose an operator $A$ such that $\|A\|=K_0$ whose norm is $K_0 $ under this case.

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

MSC numbers: 47L35