### Spectral properties of square hyponormal operators

#### Abstract

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 $\sigma(T) \cap (-\sigma(T)) \subset \{ 0 \}$,

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 $x,y \in \h$ are corresponding eigen-vectors, respectively, then $\langle \, x , \, y \rangle = 0$. Also, we define $n$th hyponormal operators and present some

properties of this kind of operators.

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