Notes on the Krupa-Zawisza Ultrapower of Self-Adjoint Operators

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^{omega} of a selfadjoint operator A. In this note, we give alternative description of A^{omega} and the Hilbert space H(A) on which A^{omega} is densely defined, which provides a criterion to determine to which representing sequence (\xi_n)n of a given vector \xi in dom(A^{omega}) has the property that A^{omega}\xi = (A\xi_n)_{omega} holds.