Operator version of the best approximation problem in Hilbert C⁎C⁎-modules

Let V⊆B(H,K)V⊆B(H,K) be a Hilbert C⁎C⁎-module over a C⁎C⁎-algebra A⊆B(H)A⊆B(H), and X,Y∈VX,Y∈V. In this paper we study a problem of finding A∈B(H)A∈B(H) such that |X+YA|⩽|X+YB||X+YA|⩽|X+YB| for all B∈AB∈A. We show that such an operator exists if and only if the range of Y⁎XY⁎X is contained in the range of Y⁎YY⁎Y, and in this case it can be chosen to belong to A″A″. We also consider Hilbert C⁎C⁎-modules in which for every X and Y there is (a unique) A with the above property.