Particular formulae for the Moore-Penrose inverses of the partitioned bounded linear operators

For any two complex Hilbert spaces H and K, let BL(H,K) be the set of bounded linear operators from H to K, and H⊕K be the direct sum of H and K. Given three Hilbert spaces H1,H2,H3 and two operators A1∈BL(H1,H3), A2∈BL(H2,H3), a partitioned bounded linear operator A=(A1,A2)∈BL(H1⊕H2,H3) can be induced, where for hi∈Hi,i=1,2. In this note we study the Moore-Penrose inverse A† of such a partitioned bounded linear operator A, and generalize a recent result of J. K. Baksalary and O. M. Baksalary [Particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix, Linear Algebra Appl. 421(2007) 16–23] from finite matrices to Hilbert space operators.