  Fuglede-Putnam theorem is not true in general for $ EP $ operators on Hilbert
spaces. We prove that under some conditions the theorem holds good. If the
adjoint operation is replaced by Moore-Penrose inverse in the theorem, we get
Fuglede-Putnam type theorem for $ EP $ operators -- however proofs are totally
different. Finally, interesting results on $ EP $ operators have been proved
using several versions of Fuglede-Putnam type theorems for $ EP $ operators on
Hilbert spaces.
