Strongly Reflexive Lattices
Strongly Reflexive Lattices
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DOI：
10.1112/jlms/s2-11.4.491
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发表时间：
1974-09
影响因子：
1.2
通讯作者：
BY W. E. Longstaff
中科院分区：
文献类型：
--
作者：
BY W. E. Longstaff
1.引言。如果IF是(复)Hubert空间，F是H的任何(闭线性)子空间的集合，则F被用来表示H上的使F的每个成员保持不变的(有界线性)算子的集合。对偶，如果9t是H上的一组算子，则用Lat 9t表示H的子空间被91的每个成员所不变的集合。子空间的集合F称为自反的，如果^=LatAlg^。如果F是自反的，则很容易证明IF是一个完备的子格，包含H的所有子空间的格的零子空间(0)和H。实际上，如果F是自反的，则它是强闭的。很容易看出，如果一个子集Mçalg F具有^=Lat 0t的性质，则给定子空间的集合F将是自反的。在J^是完全套的情况下，代数F的一阶算子集是这样的子集[4]。在F是完全原子布尔代数[2]或有限分配格[3]的情况下，也可以找到这样的子集。因此，完备套、完备原子布尔代数以及子空间的有限分配格都是自反子空间格的例子。在这篇文章中，我们指出这些结果是如何从一个相同的抽象格理论得出的。这种强自反格的理论引出了自反子空间格的新例子。它的发展和动机是这样一个问题：子空间F的哪些格是由F的一阶算子的集合0t决定的，即F=L m。作者要感谢W·H·科尼什博士在准备本论文期间进行了有益的交谈，并感谢P·R·哈尔莫斯教授对本文的发表发表了评论。
1. Introduction. If if is a (complex) Hubert space and F is any collection of (closed linear) subspaces of H 9 Alg F is used to denote the set of (bounded linear) operators on H that leave every member of F invariant. Dually, if 9t is a set of operators on H, Lat 9t is used to denote the collection of subspaces of H left invariant by every member of 91. The collection F of subspaces is called reflexive if ^=LatAlg^. If F is reflexive it is easy to show that IF is a complete sublattice, containing both the zero subspace (0) and H, of the lattice of all subspaces of H. In fact, if F is reflexive it is strongly closed [2]. It is easily seen that a given collection F of subspaces will be reflexive if a subset M ç Alg F can be found with the property that ^=Lat 0t. In the case that J^ is a complete nest, the set of operators of rank one of Alg F is such a subset [4]. Such a subset can also be found in the case that F is either a complete atomic Boolean algebra [2] or a finite distributive lattice [3]. Thus complete nests, complete atomic Boolean algebras and finite distributive lattices of subspaces are examples of reflexive subspace lattices. In this paper we indicate how these results follow from one and the same theory of abstract lattices. This theory of strongly reflexive lattices leads to new examples of reflexive subspace lattices. It develops from and is motivated by the question: Which lattices of subspaces F are determined by the set 0t of operators of rank one of Alg F in the sense that F=L Ml The author wishes to thank Dr. W. H. Cornish for helpful conversations during the preparation of this paper and Professor P. R. Halmos for comments on its presentation.