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An extension of von Neumann’s characterization of essentially selfadjoint operators is given among not necessarily densely defined symmetric operators which are assumed to be closable. Our arguments are of algebraic nature and follow the idea of [1].

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The purpose of this paper is to revise von Neumann’s characterizations of selfadjoint operators among symmetric ones. In fact, we do not assume that the underlying Hilbert space is complex, nor that the corresponding operator is densely defined, moreover, that it is closed. Following Arens, we employ algebraic arguments instead of the geometric approach of von Neumann using the Cayley transform.

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