On the generalized Fuglede-Putnam Theorem

Authors

  • M. H. M. Rashid
  • M. S. M. Noorani
  • A. S. Saari

DOI:

https://doi.org/10.5556/j.tkjm.39.2008.16

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^* $.

Author Biographies

M. H. M. Rashid

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan, Malaysia, 43600 UKM, Selangor, Malaysia.

M. S. M. Noorani

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan, Malaysia, 43600 UKM, Selangor, Malaysia.

A. S. Saari

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan, Malaysia, 43600 UKM, Selangor, Malaysia.

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Published

2008-09-30

How to Cite

Rashid, M. H. M., Noorani, M. S. M., & Saari, A. S. (2008). On the generalized Fuglede-Putnam Theorem. Tamkang Journal of Mathematics, 39(3), 239-246. https://doi.org/10.5556/j.tkjm.39.2008.16

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Papers