On the generalized Fuglede-Putnam Theorem

Abstract

In this paper, we prove the following assertions:

(1) If the pair of operators $ (A,B^*) $ satisfies

the Fuglede-Putnam Property and $ S\in \ker(\delta_{A,B}) $, where $ S\in \bh $, then we have

$$  \|\delta_{A,B}X+S\|\geq\|S\|.$$

(2) Suppose the pair of operators $ (A,B^*) $ satisfies the Fuglede-Putnam Property. If $ A^{2}X=XB^{2} $ and $ A^{3}X=XB^{3} $, then $ AX=XB $.

(3) Let $ A,B\in \bh $ be such that $ A,B^* $ are
 $ p $-hyponormal. Then  for any $ X\in\c_{2} $, $ AX-XB\in
  \mathcal{C}_{2} $ implies $ A^*X-XB^*\in \mathcal{C}_{2} $.

(4) Let $ T,S\in \bh $ be such that $ T $ and
$ S^* $ are quasihyponormal operators. If $ X\in\bh $ and $ TX=XS$ ,
then  $T^*X=XS^* $.