Abstract
Let H be a complex Hilbert space, and T be a linear bounded operator on H. T is called a quasihyponormal operator if there exists a strict monotone continuous function φ(t) on [0, + ∞) with φ(0)= 0 (standard function) such that φ(T~*T)-φ(TT~*) D_φ ≥ 0. T is called a complete hyponormal operator if D_≥ 0 is true for every standard function. T is called a θ-class operator if [T~* + T, T~*T] = 0. T is called a BN-class operator if [T~*T,TT~*] = 0. T is called a strong complete hyponormal operator if
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