Algebraic proofs of characterizing reverse order law for closed range operators in Hilbert spaces

We present more than 60 results, including some range inclusion results to characterize the reverse order law for the Moore–Penrose inverse of closed range Hilbert space operators. We use the basic properties of the Moore-Penrose inverse to prove the results. Some examples are also provided to illustrate failure cases of the reverse order law in an infinite-dimensional setting.