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Non-reflexive double triangles

Part of: General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

W. E. Longstaff
W. E. Longstaff
Affiliation:
Department of MathematicsUniversity of Western AustraliaNedlands, Western Australia6009
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Abstract

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A double triangle subspace lattice in a Hilbert space H is a 5-element set of subspaces of H, containing (0) and H, with each pair of non-trivial elements intersecting in (0) and spanning H. It is shown that if any pair of non-trivial elements has a closed vector sum the double triangle is both non-reflexive and non-transitive. A double triangle in H⊕H is an operator double triangle if each non-trivial elements is the graph of an operator acting on H. A sufficient condition is given for any operator double triangle to be non-reflexive.

MSC classification

Secondary: 47A15: Invariant subspaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 3 , December 1983 , pp. 349 - 356
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Douglas, R. G., ‘On majorization, factorization and range inclusion of operators in Hilbert space’, Proc. Amer. Math. Soc. 17 (1966), 413–416.CrossRefGoogle Scholar
[2]Hadwin, D. W., Longstaff, W. E. and Rosenthal, Peter, ‘Small transitive lattices’, Proc. Amer. Math. Soc. 87 (1983), 121–124.CrossRefGoogle Scholar
[3]Halmos, P. R., ‘Finite-dimensional Hilbert spaces’, Amer. Math. Monthly 77 (1970), 457–464.CrossRefGoogle Scholar
[4]Halmos, P. R., ‘Ten problems in Hilbert space’, Bull. Amer. Math. Soc. 76 (1970), 887–933.CrossRefGoogle Scholar
[5]Halmos, P. R., ‘Reflexive lattices of subspaces’, J. London Math. Soc. 4 (1971), 257–263.CrossRefGoogle Scholar
[6]Longstaff, W. E., ‘Strongly reflexive lattices’, J. London Math. Soc. 11 (1975), 491–498.CrossRefGoogle Scholar
[7]Longstaff, W. E. and Rosenthal, Peter, ‘On two questions of Halmos concerning subspace lattices’, Proc. Amer. Math. Soc. 75 (1979), 85–86.CrossRefGoogle Scholar
[8]Gábor, Szász, Introduction to lattice theory, 3rd ed. (Academic Press, New York and London, 1963).Google Scholar