ON THE CLASS OF $n$-NORMAL OPERATORS AND MOORE-PENROSE INVERSE

Abstract

Let $ T \in B(H)$ be a bounded linear operator on a complex Hilbert space $H$. For $ n\in \mathbb{N } $, an operator $ T\in B(H)$ is said to be n-normal if $ T^{n}T^{*}=T^{*}T^{n} $. In this paper we investigate a necessary and sufficient condition for the n-normality of $ ST $ and $ TS $, where $ S,T \in B(H). $ As a consequence, we generalize Kaplansky theorem for normal operators to n-normal operators. Also, In this paper, we provide new characterizations of n-normal operators by certain conditions involving powers of Moore-Penrose inverse.