In this paper, we show that for a positive operator $A$ on a Hilbert $C^*$-module $ \mathscr{E} $, the range $ \mathscr{R}(A) $ of $A$ is closed if and only if $ \mathscr{R}(A^\alpha) $ is closed for all $\alpha\in (0,1)\cup (1,+\infty)$, and this occurs if and only if $ \mathscr{R}(A)=\mathscr{R}(A^\alpha) $ for all $\alpha\in (0,1)\cup (1,+\infty)$. As an application, we prove that for an adjontable operator $A$ if $\mathscr{R}(A)$ is nonclosed, then $\dim\left(\overline{\mathscr{R}(A)}/\mathscr{R}(A)\right)=+\infty$. Finally, we show that for an adjointable operator $A$ if $ \overline{\mathscr{R}(A^*) } $ is orthogonally complemented in $ \mathscr{E} $, then under certain coditions there exists an idempotent $ C $ and a unique operator $ X$ such that $ XAX=X, AXA=CA, AX=C $ and $ XA=P_{A^*} $, where $ P_{A^*} $ is the orthogonal projection of $ \mathscr{E} $ onto $ \overline{\mathscr{R}(A^*)} $.

Keywords