Operators in finite distributive subspace lattices, I

The purpose of this paper is to settle in the negative an open problem in operator theory, which asks whether in a finite distributive subspace lattice ℒ on a Hilbert space, every finite rank operator of Alg ℒ can be written as a finite sum of rank one operators from Alg ℒ. The counter-example const...

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Bibliographic Details
Published in:Mathematical proceedings of the Cambridge Philosophical Society 1993-01, Vol.113 (1), p.141-146
Main Author: Spanoudakis, N. K.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematics
Operator theory
Sciences and techniques of general use
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Summary:The purpose of this paper is to settle in the negative an open problem in operator theory, which asks whether in a finite distributive subspace lattice ℒ on a Hilbert space, every finite rank operator of Alg ℒ can be written as a finite sum of rank one operators from Alg ℒ. The counter-example constructed is on a specific Hilbert space realization of the free distributive lattice on three generators and the operator which fails the above property has rank two.
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004100075824