Some relationships between an operator and its transform \(S_{r}(T)\)

Let \( T \in \mathcal {B}(\mathcal {H})\) $T\in \mathcal{B}(\mathcal{H})$ be a bounded linear operator on a Hilbert space \( \mathcal {H}\) $\mathcal{H}$ , and let \( T = U \vert T \vert \) $T=U|T|$ be the polar decomposition of T. For any \(r > 0\) $r>0$ , the transform \(S_{r}(T)\) $S_r(T)$ is defined by \(S_{r}(T) = U \vert T \vert ^{r} U\) $S_r(T)=U{|T|}^{r}U$ . In this paper, we discuss the transform \(S_{r}(T)\) $S_r(T)$ of some classes of operators such as p-hyponormal and rank one operators. We provide a new characterization of invertible normal operators via this transform. Afterwards, we investigate when an operator T and its transform \( S_{r}(T) \) $S_r(T)$ both have closed ranges, and show that this transform preserves the class of EP operators. Finally, we present some relationships between an EP operator T, its transform \( S_{r}(T)\) $S_r(T)$ and the Moore–Penrose inverse \( T^{+} \) $T^{+}$ .