Sub-direct sum of operators on Hilbert spaces and nonnegative Moore-Penrose inverses

The sub-direct sum, a generalization of normal sum operation for matrices was introduced by Fallat and Johnson [5]. Here, the definition of sub-direct sum is extended to operators between Hilbert spaces. Conditions for the sub-direct sum to have a nonnegative Moore-Penrose inverse are obtained when the summands themselves have nonnegative Moore-Penrose inverses. The converse problem is also considered.