Identities concerning the reverse order law for the Moore-Penrose inverse

We prove some identities related to the reverse order law for the Moore-Penrose inverse of operators on Hilbert spaces, extending some results from (Y. Tian, S. Cheng, Linear Multilinear Algebra 52 (2004)) and (R. E. Cline, SIAM Rev., Vol. 6, No. 1 (1964)) to infinite dimensional settings.