Abstract. A densely defined operator $T$ acting between Hilbert spaces is shown to be closed if and only if $T^*T$ and $TT^*$ are both selfadjoint operators on the corresponding Hilbert spaces. This is an extension of the classical von Neumann theorem [vonNeumann1930] on the selfadjointness of $T^*T$ whenever $T$ is closed.

DOI: 10.14232/actasm-013-283-x

AMS Subject Classification (1991): 47B25, 47B65

Keyword(s): Hilbert space, closed operators, selfadjoint operators, von Neumann theorem

Received May 8, 2013, and in revised form May 16, 2013. (Registered under 33/2013.)