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

Let \( T \in \mathcal {B}(\mathcal {H})\) be a bounded linear operator on a Hilbert space \( \mathcal {H}\), and let \( T = U \vert T \vert \) be the polar decomposition of T. For any \(r > 0\), the transform \(S_{r}(T)\) is defined by \(S_{r}(T) = U \vert T \vert ^{r} U\). In this paper, we discuss the transform \(S_{r}(T)\) 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 \( S_{r}(T) \) 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 \( S_{r}(T)\) and the Moore–Penrose inverse \( T^{+} \).