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