Kaufman’s representation theorem is that a closed operator S with a dense domain in Hilbert space H is represented by a quotient S = B/(1 − B*B)1/2 for a unique contraction B. When S is a symmetric operator, what is a condition of the spectrum of B to admit selfadjoint extensions of S? In this note, it is shown that if there are no negative real points in the spectrum of B, then S has selfadjoint extensions.
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 .