Modern approaches to the invariant‐subspace problem (Cambridge Tracts in Mathematics 188)
Modern approaches to the invariant‐subspace problem (Cambridge Tracts in Mathematics 188)
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DOI：
10.1112/blms/bdt016
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发表时间：
2013-08
影响因子：
0.9
通讯作者：
T. Gillespie
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文献类型：
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作者：
T. Gillespie
不变子空间问题是问在一个维数至少为2的复Banach空间X上是否每个有界线性算子T都有一个非平凡的闭不变子空间;也就是说，是否存在X的一个闭子空间M，它既不同于{0}也不同于X，使得T（M）<$M？如果X是有限维的，则T将具有特征向量，因此具有一维不变子空间，而如果X是不可分的并且0= x∈ X，则{Tnx：n= 0，1，2，.}的闭线性跨度是有限维的。是非平凡的和T不变的。因此，在这个问题的兴趣在于的情况下，当X是无限维和可分的。这个问题至少可以追溯到20世纪50年代，当时Aronszajn和Smith [1]给出了Banach空间上紧算子的正解，这一结果在希尔伯特空间的情况下已经由von Neumann在未发表的工作中解决。紧情形由伯恩斯坦和罗宾逊[2]推广，他们使用非标准分析技巧证明了如果p（T）对于某个非零多项式是紧的，则T具有非平凡的闭不变子空间。证明，适用于希尔伯特空间，在[6]中使用标准方法进行了修改，并由邦索尔[3]进一步简化。邦索尔的证明在Banach空间中是有效的。下一个里程碑是显着延长伯恩斯坦-罗宾逊定理获得于1973年由六罗蒙诺索夫使用一个全新的方法。Lomonosov的结果指出，Banach空间上的每个T，如果与非零紧算子交换，并且不是单位算子的标量倍，则有一个非平凡的闭超不变子空间（即对T的交换子不变）。Banach空间的一般不变子空间问题在[1]中明确地陈述过，作者评论说，它甚至在Hilbert空间中也是开放的。现在已知一般Banach空间有一个负解，一个反例由Enflo [4，5]给出，另一个由Read [7]给出，他还给出了经典空间上的第一个反例，即序列空间l1[8]。然而，这个问题对于希尔伯特空间，或者更一般地对于自反Banach空间仍然是开放的。多年来，它提供了刺激大量的发展算子理论，无论是在Banach空间和希尔伯特空间。本书的目的就是要说明其中的一些发展。第一章包括标准的背景材料，从功能分析，运营商理论和复杂的分析，将需要以后。讨论，然后移动到一个详细的帐户的方法，建设不变的子空间所产生的开创性的结果获得SW布朗在1978年断言，每一个次正常运营商的希尔伯特空间有一个非平凡的封闭不变的子空间。希尔伯特空间H上的算子称为次正规的，如果它具有作用在包含H的希尔伯特空间上的正规算子的扩张。从谱定理可知，正规算子有大量的不变子空间，自然的下一步就是考察次正规的情形。Brown建立的关键结果是，如果T是具有循环向量的非正规次正规算子，则在基础Hilbert空间中存在向量x和y以及复数λ，使得p（λ）=<$p（T）x，y <$
The invariant-subspace problem asks whether every bounded linear operator T on a complex Banach space X of dimension at least 2 has a non-trivial closed invariant subspace; that is, does there exist a closed subspace M of X, different from both {0} and X, such that T (M)⊆ M? If X is finite-dimensional, then T will have an eigenvector and hence a onedimensional invariant subspace whilst, if X is non-separable and 0= x∈ X, then the closed linear span of {Tnx: n= 0, 1, 2,...} is non-trivial and T-invariant. Thus the interest in the problem lies in the case when X is infinite-dimensional and separable. The problem dates back at least as far as the 1950s when Aronszajn and Smith [1] gave a positive solution for compact operators on Banach spaces, the Hilbert space case of this result having been settled earlier by von Neumann in unpublished work. The compact case was extended by Bernstein and Robinson [2], who showed using non-standard analysis techniques that T has a non-trivial closed invariant subspace if p (T) is compact for some non-zero polynomial. The proof, which applied to Hilbert spaces, was reworked in [6] using standard methods and further simplified by Bonsall [3]. Bonsall’s proof is valid in a Banach space setting. The next milestone was the significant extension of the Bernstein–Robinson theorem obtained in 1973 by VI Lomonosov using an entirely new approach. The Lomonosov result states that every T on a Banach space that commutes with a non-zero compact operator and is not a scalar multiple of the identity operator has a non-trivial closed hyperinvariant subspace (that is, invariant for the commutant of T).The general invariant-subspace problem for Banach spaces was stated explicitly in [1], the authors commenting that it was open even in the Hilbert space setting. It is now known to have a negative solution for a general Banach space, a counterexample having been given by Enflo [4, 5] and another by Read [7], who also gave the first counterexample on a classical space, namely on the sequence space l1[8]. However, the problem remains open for Hilbert space, or more generally for reflexive Banach spaces. Over the years, it has provided the stimulus for a great number of developments in operator theory, both on Banach and on Hilbert spaces. The aim of the present book is to give an account of some of these developments. The first chapter comprises standard background material from functional analysis, operator theory and complex analysis that will be needed later. The discussion then moves on to a detailed account of a method for constructing invariant subspaces that arose from the seminal result obtained by SW Brown in 1978 asserting that every subnormal operator on a Hilbert space has a non-trivial closed invariant subspace. An operator on a Hilbert space H is said to be subnormal if it has an extension to a normal operator acting on a Hilbert space containing H. Given that normal operators have a wealth of invariant subspaces from the spectral theorem, it was a natural next step to examine the subnormal case. The key result that Brown established was that, if T is a non-normal subnormal operator with a cyclic vector, then there exist vectors x and y in the underlying Hilbert space and a complex number λ such that p (λ)=〈 p (T) x, y〉