On the semi-hyponormal operators on a Hilbert space
Commun. Korean Math. Soc. 1997 Vol. 12, No. 3, 597-602
Hyung Koo Cha
Hanyang University
Abstract : Let $\cHc$ be a separable complex Hilbert space and $\LH$ be the $*$-algebra of all bounded linear operators on $\cHc$. For $T\in\LH$, we construct a pair of semi-positive definite operators $$|T|_r=(T^*T)^{\frac 12}\quad\text{and}\quad |T|_l=(TT^*)^{\frac 12}.$$ An operator $T$ is called a semi-hyponormal operator if $$Q_T=|T|_r - |T|_l \ge 0.$$ In this paper, by using a technique introduced by Berberian [1], we show that the approximate point spectrum $\sigma_{ap}(T)$ of a semi-hyponormal operator T is empty.
Keywords : polar decomposition, semi-hyponormal, spectrum, approximate point spectrum, faithful $*$-representation, irreducible, pure
MSC numbers : 47A10, 47A15, 47A67, 47B20
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