Operator version of the best approximation problem in Hilbert -modules

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