Notes on the Krupa-Zawisza ultrapower of self-adjoint operators

• Autorzy: Ando H. , Saigo H. , Ojima I.
• Języki publikacji: EN

Abstrakty

EN

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_ωξ = (Aξ_n)_ω holds. An explicit core for A_ω is also described.

Słowa kluczowe

EN

ultraproduct; unbounded self-adjoint operators

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2014
• Tom: Vol. 34, Fasc. 1
• Strony: 147--159
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Ando H.

• Institut des Hautes Études Scientifiques, Le Bois-Marie 35, route de Chartres, 91440 Bures-sur-Yvette, France

autor

Saigo H.

• Nagahama Institute of Bio-Science and Technology, Nagahama 526-0829, Japan

autor

Ojima I.

• Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kitashirakawa Oiwakecho, 606-8502 Kyoto, Japan

Bibliografia

• [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.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).