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Contents of PMS, Vol. 34, Fasc. 1,
pages 147 - 159
 

NOTES ON THE KRUPA–ZAWISZA ULTRAPOWER OF SELF-ADJOINT OPERATORS

Hiroshi Ando
Izumi Ojima
Hayato Saigo

Abstract: Let ω ∈ βℕ\ ℕ  be a free ultrafilter on ℕ  . It is known that there is a difficulty in constructing the ultrapower of unbounded operators. Krupa and Zawisza gave a rigorous definition of the ultrapower A
  ω  of a self-adjoint operator A . In this note, we give an alternative description of A
  ω  and the Hilbert space H (A ) on which A
  ω  is densely defined. This provides a criterion to determine a representing sequence (ξ )
  n n  of a given vector ξ ∈ dom (A )
         ω which has the property that A  ξ = (A ξ)
  ω      n ω  holds. An explicit core for A
  ω  is also described.

2000 AMS Mathematics Subject Classification: Primary: 47A10; Secondary: 03C20.

Keywords and phrases: Ultraproduct, unbounded self-adjoint operators.

References

[1]   H. Ando and U. Haagerup, Ultraproducts of von Neumann algebras, arXiv: 12112.5457v2.

[2]   A. Kishimoto, Rohlin property for flows, Contemp. Math. 335 (2003), pp. 195–207.

[3]   A. Krupa and B. Zawisza, Applications of ultrapowers in analysis of unbounded selfadjoint operators, Bull. Polish Acad. Sci. Math. 32 (1984), pp. 581–588.

[4]   A. Krupa and B. Zawisza, Ultrapowers of unbounded selfadjoint operators, Studia Math. 87 (2) (1987), pp. 101–120.

[5]   T. Masuda and R. Tomatsu, Rohlin flows on von Neumann algebras, preprint, arXiv: 1206.0955v2.

[6]   H. Saigo, Categorical non-standard analysis, preprint, arXiv: 1009.0234.

[7]   K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Grad. Texts in Math., Springer, 2012.

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