Fuglede-Putnam type commutativity theorems for $EP$ operators

Abstract: 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.