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.