The Present State and Heritages of the Invariant Subspace Problem
The Present State and Heritages of the Invariant Subspace Problem
复制标题
不变子空间问题的现状和遗产
DOI：
10.1007/s00032-005-0048-7
复制
发表时间：
2005
影响因子：
1.7
通讯作者：
B. S. Yadav
中科院分区：
文献类型：
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作者：
B. S. Yadav
在算符理论中有一个突出的问题，即所谓的“不变子空间问题”，这个问题已经提出了半个多世纪。尽管功能分析人员做出了巨大的努力，但直到今天，这个问题仍然困扰着他们。有时在间隔十多年之后取得了重大成就，但其解决办法似乎遥遥无期。这个问题的历史错综复杂，交织着希望和失望，但就目前而言，分析师们似乎没有足够犀利的工具来解决这个问题。本文的目的是简要讨论，一方面，历史上为解决这个问题所做的各种尝试；另一方面，我们将设法描述在寻求其解决办法的过程中所收集到的有关其遗产的宝藏。我们的想法是吸引那些可能不精通泛函分析和算子理论，但对了解不变子空间问题的当前分支感兴趣的人的注意，而不会对他们的放纵造成太多负担。与此同时，我们试图提供一个即使是专家也会津津乐道的主题概述。这个问题。设X是一个巴纳赫空间，B (X)是X上所有有界线性算子的巴纳赫代数。如果T (M)∧M，则X的闭子空间M称为算子T∈B (X)的不变子空间。由于{o}和X在T下是平凡不变的，如果M={o}，则M称为非平凡的，如果X在T下是不变的，则M称为T的超不变子空间
There is an outstanding problem in operator theory, the so-called “Invariant Subspace Problem” which has been open for more than half a century. In spite of momentous efforts by functional analysts, the problem continues to elude them even today. There have been significant achievements on occasions, sometimes after an interval of more than a decade in between, but its solution seems to be nowhere in sight. The problem has a chequered history interwoven with hopes and disappointments, but for the moment, it appears that analysts have no tools in their kit sharp enough even to attack the problem. The object of this article is to discuss in brief, on one hand, a history of various attempts made to solve the problem; on the other, we shall try to describe the treasure trove of its heritages that have gathered around while traversing the quest for its solution. Our idea is to capture the attention of those who may not be well-versed in functional analysis and operator theory but are interested in knowing the present ramifications of the invariant subspace problem without taxing much on their indulgence. At the same time, we attempt to provide an overview of the subject which even a specialist would relish.The Problem. Let X be a Banach space and B (X) the Banach algebra of all bounded linear operators on X. A closed subspace M of X is called an invariant subspace of an operator T∈ B (X) if T (M)⊂ M. As {o} and X are trivially invariant under T, M is called non-trivial if M={o}, X. M is called a hyperinvariant subspace of T if it is invariant under every