Hilberträume mit amorphen Basen. (German) Zbl 0554.47001

Brunner, Norbert

Compos. Math. 52, 381-387 (1984).

Ramsey’s theorem RT is the assertion, that for every infinite set X, every set \(C\subseteq [X]^ 2\) (the family of pairs) there is an infinite \(H\subseteq X\) such that \([H]^ 2\subseteq C\) or \([H]^ 2\cap C=\emptyset.\) RT is a weak form of the axiom of choice which is consistent with the existence of an amorphous set (an infinite set each of its infinite subsets is cofinite). RT implies, that every operator on a Hilbert space with an amorphous base is a direct sum of a finite matrix and a scalar operator. RT is necessary to obtain this conclusion.

MSC:

47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
03E25	Axiom of choice and related propositions
46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords:

Ramsey’s theorem; weak form of the axiom of choice; existence of an amorphous set; every operator on a Hilbert space with an amorphous base is a direct sum of a finite matrix and a scalar operator

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