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

TL;DR

This paper offers an alternative description of the ultrapower of self-adjoint operators and the associated Hilbert space, providing a criterion for representing sequences that ensure the ultrapower acts as expected.

Contribution

It introduces a new characterization of the ultrapower of self-adjoint operators and the Hilbert space where it is densely defined, improving understanding of their structure.

Findings

Provides an alternative description of A^{omega}

Establishes a criterion for representing sequences

Clarifies the domain and action of the ultrapower

Abstract

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.