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

Ando H., Saigo H., Ojima I.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 1

147--159

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 Aω of a self-adjoint operator A. In this note, we give an alternative description of Aω and the Hilbert space H(A) on which Aω is densely defined. This provides a criterion to determine a representing sequence (ξn)n of a given vector ξ ∈ dom(Aω) which has the property that Aωξ = (Aξn)ω holds. An explicit core for Aω is also described.