Asymmetric Fuglede Putnam's Theorem for operators reduced by their   eigenspaces
  Fuglede-Putnam Theorem have been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators. 
Farida Lombarkia and Mohamed Amouch
math.FA