Interpolation in self-adjoint settings

Abstract:

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