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WROCŁAW UNIVERSITY
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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.

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