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ON SMALL SUBSPACE LATTICES IN HILBERT SPACE

Part of: General theory of linear operators Linear spaces and algebras of operators

Published online by Cambridge University Press:  15 October 2013

AIJU DONG ,
WENMING WU  and
WEI YUAN
AIJU DONG
Affiliation:
College of Mathematics and Computer Engineering, Xi’an University of Arts and Science, Xi’an 710065, PR China email daj1965@163.com
WENMING WU*
Affiliation:
College of Mathematics, Chongqing Normal University, Chongqing 400047, PR China
WEI YUAN
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100190, PR China email wyuan@math.ac.cn
*
✉wuwm@amss.ac.cn
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Abstract

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We study the reflexivity and transitivity of a double triangle lattice of subspaces in a Hilbert space. We show that the double triangle lattice is neither reflexive nor transitive when some invertibility condition is satisfied (by the restriction of a projection under another). In this case, we show that the reflexive lattice determined by the double triangle lattice contains infinitely many projections, which partially answers a problem of Halmos on small lattices of subspaces in Hilbert spaces.

Keywords

projectionslatticedouble trianglereflexivetransitive

MSC classification

Secondary: 47A62: Equations involving linear operators, with operator unknowns 47L75: Other nonselfadjoint operator algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 1 , February 2014 , pp. 44 - 60
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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