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Abstract  

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.

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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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operators and p -adic quantization, Acta Appl. Math. 57 (1999), 205-237. MR 2001i :47117 Non-Archimedean analogues of orthogonal and symmetric operators and p-adic quantization Acta Appl. Math

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References [1] Ando , T. Nishio , K. 1970 Positive selfadjoint extensions of positive symmetric operators

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References [1] Ando , T. Nishio , K. 1970 Positive selfadjoint extensions of positive symmetric operators

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.N., Extensions of symmetric operators and symmetric binary relations, Mat. Zametki , 17 (1975), 41–48; English transl. in Math. Notes , 17 (1975), 25–28. Kochubei A.N. Extensions of

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the product of two symmetric operator matrices and its application in elasticity, Chinese Phys. B, 20, 124601. doi:10.1088/1674-1056/20/12/124601 [36] Kac , V. and Cheung

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