Moore–Penrose inverses of partitioned adjointable operators on Hilbert C∗-modules

Let A be a C∗-algebra, Hi(i=1,2,3) be three Hilbert-A modules, A1∈L(H1,H3) and A2∈L(H2,H3), where L(H1,H3) (resp. L(H2,H3)) is the set of the adjointable operators from H1 to H3 (resp. H2 to H3). For such two operators A1 and A2, a 1×2 partitioned operator A=(A1,A2) can be induced by letting for hi∈Hi,i=1,2. In this paper, several formulae for the Moore–Penrose inverse A† of A are derived, and an approach to constructing the weighted Moore–Penrose inverse from the non-weighted case is provided. In particular, the main result of Udwadia and Phohomsiri [F.E. Udwadia, P. Phohomsiri, Recursive formulas for the generalized LM-inverse of a matrix, J. Optimiz. Theory App. 131 (2006) 1–16] is generalized.