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A Hilbert space operator T satisfying either (**) | (TV, ?) 2<g (| T\£, c) (| T\ r, , -/;) for all <=, ;j e^ , or (*) |(7V-, £)|^(|T|£, =) for all ^^M is studied. The condition (**) defines a slightly larger class than the hyponormality, and for compact operators (**) is equivalent to the normality. The condition (*) is characterized by using an operator whose numerical radius is less than 1, and among other things we show that (*) and the normality are equivalent for matrices. Moreover, we show that (*) and the normality are equivalent for trace class operators in Appendix.

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