Thai Journal of Mathematics Volume 8 (2010) Number 1 : 185–192 www.math.science.cmu.ac.th/thaijournal Online ISSN 1686-0209 Some Conditions on Non-Normal Operators which Imply Normality M.H.M. Rashid Abstract : In this paper, we prove the following assertions: (i) Let A,B,X ∈ B(H) be such that A * is p-hyponormal or log-hyponormal, B is a dominant and X is invertible. If XA = BX, then there is a unitary operator U such that AU = UB and hence A and B are normal. (ii) Let T = A + iB ∈ B(H) be the cartesian decomposition of T with AB is p-hyponormal. If A or B is positive, then T is normal. (iii) Let A,V,X ∈ B(H) be such that V,X are isometries and A * is p-hyponormal. If VX = XA, then A is unitary. (iv) Let A, B ∈ B(H) be such that A + B ≥±X. Then for every paranormal operator X ∈ B(H) we have ‖AX + XB‖≥‖X‖ 2 . Keywords : Hyponormal operators, Fuglede-Putnam Theorem, and Commuta- tor. 2000 Mathematics Subject Classification : 47A10, 47B20 1 Introduction Let H be infinite dimensional complex Hilbert, and let B(H) denote the alge- bra of all bounded linear operators acts on H . Let ‖.‖ denote the spectral norm, and 〈., .〉 be an inner product in H. For T ∈ B(H), we denote the spectrum and the point spectrum of T by σ(T ), σ p (T ). An operator A ∈ B(H) is called positive if 〈Ax, x〉≥ 0 for all non-zero vectors x ∈H , isometry if ‖Ax‖ = ‖x‖ for all a non-zero vector x ∈H , unitary if Copyright c  2010 by the Mathematical Association of Thailand. All rights reserved.