Moore–Penrose inverses of partitioned adjointable operators on Hilbert [formula omitted]-modules

Abstract

Let A be a C ∗ -algebra, H i ( i = 1 , 2 , 3 ) be three Hilbert- A modules, A 1 ∈ L ( H 1 , H 3 ) and A 2 ∈ L ( H 2 , H 3 ) , where L ( H 1 , H 3 ) (resp. L ( H 2 , H 3 ) ) is the set of the adjointable operators from H 1 to H 3 (resp. H 2 to H 3 ). For such two operators A 1 and A 2 , a 1 × 2 partitioned operator A = ( A 1 , A 2 ) can be induced by letting A h 1 h 2 = A 1 h 1 + A 2 h 2 for h i ∈ H i , 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.