Abstract. We characterize those subpositive operators for which their Krein--von Neumann extension has closed range, moreover we construct their Moore--Penrose inverse. Our treatment follows as a tool the factorization approach to the extension theory of positive operators. As addition we give a short proof of Dixmier's theorem that a bounded positive operator $A$ and its square root $A^{1/2}$ have the same range if and only if $A$ has closed range and of Banach's closed range theorem for Hilbert space operators.

AMS Subject Classification (1991): 47A20, 47B65, 47A05

Keyword(s): characterization, positive operator, closed range, Krein--von Neumann extension, Moore--Penrose inverse

Received December 23, 2009, and in revised form February 1, 2011. (Registered under 6468/2009.)