Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$ C ∗ -modules

Abstract

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 $$ \overline{\mathscr {R}(A^*)}$$ .