Let $mathcal{L}$ be a subspace lattice on a Hilbert space $mathcal{H}$. Let x and y be vectors in $mathcal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE$ = $EP_x$ for each E ${in};mathcal{L}$, then the following are equivalent. (1) There exists an operator A in Alg$mathcal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{perp}$ and A ${geq}$ 0. (2) sup ${frac{{parallel}E^{perp}y{parallel}}{{parallel}E^{perp}x{parallel}}:E{in}mathcal{L}}$ < ${infty}$ < x, y > ${geq}$ 0. Let X and Y be operators in $mathcal{B}(mathcal{H})$. Let P be the projection onto $overline{rangeX}$. If PE = EP for each E ${in};mathcal{L}$, then the following are equivalent: (1) sup ${frac{{parallel}E^{perp}Yf{parallel}}{{parallel}E^{perp}Xf{parallel}}:f{in}mathcal{H},E{in}mathcal{L}}$ < ${infty}$ and < Xf, Yf > ${geq}$ 0 for all f in H. (2) There exists a positive operator A in Alg$mathcal{L}$ such that AX = Y.