Characterization of subprojection suboperators (Q803907)

Following \textit{P. R. Halmos} [Anniversary volume on approximation theory and functional analysis, ISNM 65, 49-65 (1984; Zbl 0557.47017)] by a suboperator one means a bounded linear transformation from a subspace \(H_ 0\) of a Hilbert space H into the whole space H. A suboperator is called a subpositive or a subprojection suboperator if it extends to a positive operator, respectively to a projection on H. The main result of the paper asserts that a suboperator Q: \(H_ 0\to H\) is a subprojection if and only if \(\| Qx\|^ 2=(Qx,x)\) for all x in \(H_ 0\). Using this result the author gives a new proof for the characterization obtained by Halmos for subprojection or subpositive suboperators. Factorizations through projections are characterized also.