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 ,..., D n } in the commutant, and all collections of vectors {f 1 , f 2 ,..., f n }, we have ∥Σ n j=1 D j Yf j ∥ ≤ K ∥ Σ n j=1 D j X f j ∥. We also study the equality ∥DY f ∥ = K∥DX f ∥, in connection with solving the equation AX = Y where the operator A is required to lie in some CSL algebra.