Separated Pairs of Submodules in Hilbert C^*-modules

We introduce the notion of the separated pair of closed submodules in the setting of Hilbert C^*-modules. We demonstrate that even in the case of Hilbert spaces this concept has several nice characterizations enriching the theory of separated pairs of subspaces in Hilbert spaces. Let H and K be orthogonally complemented closed submodules of a Hilbert C^*-module E. We establish that (H, K) is a separated pair in E if and only if there are idempotents ₁ and ₂ such that ₁₂=₂₁=0 and R (₁) = H and R (₂) = K. We show that R (₁+₂) is closed for each C if and only if R (₁+₂) is closed. We use the localization of Hilbert C^*-modules to define the angle between closed submodules. We prove that if (H^, K^) is concordant, then (H^, K^) is a separated pair if the cosine of this angle is less than one. We also present some surprising examples to illustrate our results.