Interpolation in self-adjoint settings

We study the operator equation A X = Y AX=Y , where the operators X X and Y Y are given and the operator A A is required to lie in some von Neumann algebra. We derive a necessary and sufficient condition for the existence of a solution A A . The condition is that there must exist a constant K K so that, for all finite collections of operators { D 1 , D 2 , … , D n } \{D_{1},D_{2}, \dots , D_{n}\} in the commutant, and all collections of vectors { f 1 , f 2 , … , f n } \{f_{1}, f_{2}, \dots , f_{n}\} , we have ‖ ∑ j = 1 n D j Y f j ‖ ≤ K ‖ ∑ j = 1 n D j X f j ‖ . \Vert \sum _{j=1}^{n} D_{j} Y f_{j} \Vert \leq K\, \Vert \sum _{j=1}^{n} D_{j} X f_{j} \Vert \;. We also study the equality ‖ D Y f ‖ = K ‖ D X f ‖ \Vert DYf\Vert = K\Vert DXf\Vert , in connection with solving the equation A X = Y AX=Y where the operator A A is required to lie in some CSL algebra.