Operators in finite distributive subspace lattices II
Spanoudakis, N.
Studia Mathematica, Tome 108 (1994), p. 223-239 / Harvested from The Polish Digital Mathematics Library
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In a previous paper we gave an example of a finite distributive subspace lattice ℒ on a Hilbert space and a rank two operator of Algℒ that cannot be written as a finite sum of rank one operators from Algℒ. The lattice ℒ was a specific realization of the free distributive lattice on three generators. In the present paper, which is a sequel to the aforementioned one, we study Algℒ for the general free distributive lattice with three generators (on a normed space). Necessary and sufficient conditions are given for 1) a finite rank operator of Algℒ to be written as a finite sum of rank ones from Algℒ, and 2) a realization of ℒ to contain a finite rank operator of Algℒ with the preceding property. These results are then used to show the curiosity that the product of two finite rank operators of Algℒ always has the above property.

Publié le : 1994-01-01
EUDML-ID : urn:eudml:doc:216130 
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Spanoudakis, N. Operators in finite distributive subspace lattices II. Studia Mathematica, Tome 108 (1994) pp. 223-239. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv111i3p223bwm/

[00000] [1] J. A. Erdos, Operators of finite rank in nest algebras, J. London Math. Soc. 43 (1968), 391-397. | Zbl 0169.17501

[00001] [2] A. Hopenwasser and R. Moore, Finite rank operators in reflexive operator algebras, J. London Math. Soc. (2) 27 (1983), 331-338. | Zbl 0488.47004

[00002] [3] M. S. Lambrou, Approximants, commutants and double commutants in normed algebras, ibid. 25 (1982), 499-512. | Zbl 0457.47009

[00003] [4] W. E. Longstaff, Strongly reflexive lattices, ibid. 11 (1975), 491-498. | Zbl 0313.47002

[00004] [5] W. E. Longstaff, Operators of rank one in reflexive algebras, Canad. J. Math. 28 (1976), 19-23. | Zbl 0317.46052

[00005] [6] N. K. Spanoudakis, Generalizations of certain nest algebra results, Proc. Amer. Math. Soc. 115 (1992), 711-723. | Zbl 0781.47016

[00006] [7] N. K. Spanoudakis, Operators in finite distributive subspace lattices I, Math. Proc. Cambridge Philos. Soc. 113 (1993), 141-146.