Operators in finite distributive subspace lattices II

N. Spanoudakis

Studia Mathematica (1994)

  • Volume: 111, Issue: 3, page 223-239
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Spanoudakis, N.. "Operators in finite distributive subspace lattices II." Studia Mathematica 111.3 (1994): 223-239. <http://eudml.org/doc/216130>.

@article{Spanoudakis1994,
abstract = {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.},
author = {Spanoudakis, N.},
journal = {Studia Mathematica},
keywords = {finite distributive subspace lattice; free distributive lattice with three generators; finite rank operator; finite sum of rank},
language = {eng},
number = {3},
pages = {223-239},
title = {Operators in finite distributive subspace lattices II},
url = {http://eudml.org/doc/216130},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Spanoudakis, N.
TI - Operators in finite distributive subspace lattices II
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 3
SP - 223
EP - 239
AB - 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.
LA - eng
KW - finite distributive subspace lattice; free distributive lattice with three generators; finite rank operator; finite sum of rank
UR - http://eudml.org/doc/216130
ER -

References

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  1. [1] J. A. Erdos, Operators of finite rank in nest algebras, J. London Math. Soc. 43 (1968), 391-397. Zbl0169.17501
  2. [2] A. Hopenwasser and R. Moore, Finite rank operators in reflexive operator algebras, J. London Math. Soc. (2) 27 (1983), 331-338. Zbl0488.47004
  3. [3] M. S. Lambrou, Approximants, commutants and double commutants in normed algebras, ibid. 25 (1982), 499-512. Zbl0457.47009
  4. [4] W. E. Longstaff, Strongly reflexive lattices, ibid. 11 (1975), 491-498. Zbl0313.47002
  5. [5] W. E. Longstaff, Operators of rank one in reflexive algebras, Canad. J. Math. 28 (1976), 19-23. Zbl0317.46052
  6. [6] N. K. Spanoudakis, Generalizations of certain nest algebra results, Proc. Amer. Math. Soc. 115 (1992), 711-723. Zbl0781.47016
  7. [7] N. K. Spanoudakis, Operators in finite distributive subspace lattices I, Math. Proc. Cambridge Philos. Soc. 113 (1993), 141-146. 

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