Let $mathcal{L}$ be a subspace lattice on a Hilbert space $mathcal{H}$. And let X and Y be operators acting on a Hilbert space $mathcal{H}$. Let $sp(x)={{alpha}x;:;{alpha}{in}mathcal{C}}$ $x{in}mathcal{H}$. Assume that $mathcal{H}=overline{range;X}{oplus}sp(h)$ for some $h{in}mathcal{H}$ and < $h$, $E^{ot}Xf$ >= 0 for each $f{in}mathcal{H}$ and $E{in}mathcal{L}$. Then there exists an operator A in Alg$mathcal{L}$ such that AX = Y if and only if $sup{frac{{parallel}E^{ot}Yf{parallel}}{{parallel}E^{ot}Yf{parallel}};:;f{in}H,;E{in}mathcal{L}}$ = K < ${infty}$. Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and ${parallel}||A{parallel}=K$.