In this paper, we introduce a square hyponormal operator as a bounded linear operator T on a complex Hilbert space H such that T 2 is a hyponormal operator, and we investigate some basic properties of this operator. Under the hypothesis σ(T) ∩ (−σ(T)) ⊂ {0}, we study spectral properties of a square hyponormal operator. In particular, we show that if z and w are distinct eigen-values of T and x, y ∈ H are corresponding eigen-vectors, respectively, then x, y = 0. Also, we define nth hyponormal operators and present some properties of this kind of operators