Communications of the
Korean Mathematical Society CKMS

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