Closed range and nonclosed range adjointable operators on Hilbert -modules

摘要

In this paper, we show that for a positive operator A on a Hilbert -module , the range of A is closed if and only if is closed for all , and this occurs if and only if for all . As an application, we prove that for an adjontable operator A if is nonclosed, then . Finally, we show that for an adjointable operator A if is orthogonally complemented in , then under certain coditions there exists an idempotent C and a unique operator X such that and , where is the orthogonal projection of onto (R (A*))over bar.