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Operators in finite distributive subspace lattices, I

Published online by Cambridge University Press:  24 October 2008

N. K. Spanoudakis
N. K. Spanoudakis
Affiliation:
Department of Mathematics, University of Crete, 714 09Iraklio, Crete, Greece
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 1 , January 1993 , pp. 141 - 146
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Argyros, S., Lambrou, M. and Longstaff, W. E.. Atomic Boolean Subspace Lattices and Applications to the Theory of Bases. Memoirs Amer. Math. Soc. no. 445 (American Mathematical Society, 1991).CrossRefGoogle Scholar
[2]Berkhoff, G.. Lattice Theory, third edition. Amer. Math. Soc. Colloq. Publ. vol. 25 (American Mathematical Society, 1967).Google Scholar
[3]Erdos, J. A.. Operators of finite rank in nest algebras. J. London Math. Soc. 43 (1968), 391–397.CrossRefGoogle Scholar
[4]Hopenwasser, A. and Moore, R.. Finite rank operators in reflexive operator algebras. J. London Math. Soc. (2) 27 (1983), 331–338.CrossRefGoogle Scholar
[5]Lambrou, M. S.. Approximants, commutants and double commutants in normed algebras. J. London Math. Soc. (2) 25 (1982), 499–512.CrossRefGoogle Scholar
[6]Lambrou, M. S. and Longstaff, W. E.. Unit ball density and the operator equation AX = YB. J. Operator Theory, to appear.Google Scholar
[7]Longstaff, W. E.. Strongly reflexive lattices. J. London Math. Soc. (2) 11 (1975), 491–498.CrossRefGoogle Scholar
[8]Longstaff, W. E.. Operators of rank one in reflexive algebras. Canad. J. Math. 28 (1976), 19–23.CrossRefGoogle Scholar
[9]Spanoudakis, N. K.. Generalizations of certain nest algebra results. Proc. Amer. Math. Soc. 115 (1992), 711–723.CrossRefGoogle Scholar
[10]Spanoudakis, N. K.. Operators in finite distributive subspaee lattices, II. Preprint (1992).Google Scholar