Hilbert空间上算子逼近问题

Since Halmos’ famous article “Ten problems in Hilbert space”, there have been a lot of work concerning on approximation problems. Apostol, Fialkow, Herrero, Voiculescu and other experts deal with approximation problems related to some subsets of L(H) that are invariant under similarity. Many work focus on the following problem. Given a subset R of L(H) invariant under similarity, defined in algebraic, geometric or analytic term, characterize its norm-closure in simple terms. The closures of the set of algebraic operators, the set of nilpotent operators and the set of polynomially compact operators are determined by using different subsets of the spectra and index function of the operators. We shall consider other approximation problems of Hilbert space operators. We shall characterize those operators which has an arbitrarily small compact perturbation to satisfy some fixed property, such as Coburn type properties, mixing type properties and power regularity. Further we will consider stability of such properties under small compact perturbations. On the other hand, we shall consider almost anticommuting self-adjoint matrices problem and give a quantitative estimate of such problems.

自从Halmos的著名文章“Ten problems in Hilbert space”发表之后，关于算子逼近方面问题产生许多重要工作。Apostol，Fialkow，Herrero，Voiculescu和其他学者针对L(H)的某些相似不变子集展开逼近问题研究。给定L(H)的某个子集R，它可能是以代数，几何或分析的方式定义的，利用某些简单的语言刻画R的范数闭包。例如利用谱的不同部分和指标函数的语言可以刻画全体代数算子，全体幂零算子以及全体多项式紧算子的范数闭包。我们将进一步考虑Hilbert空间上算子的逼近问题。我们将刻画何时算子可以通过任意小的紧扰动满足某种特定性质，如Coburn型性质，mixing型性质以及幂正则性。进一步，我们将考虑几乎反交换自伴矩阵的逼近问题，尝试给出量化估计。

我们在Hilbert空间以及算子代数框架下研究了逼近问题。实际上，我们针对Hilbert空间算子定义了Coburn型性质、EP类算子、性质P以及共轭对称性。我们针对这些性质研究了相关逼近问题。我们刻画了哪些算子可以通过任意小的紧扰动变为具有其中某种性质的算子。同时，我们还研究了这些性质在小的紧扰动下的稳定性。我们关于Coburn型性质研究的文章发表于Chinese Ann. Math. Ser. B杂志。其他的文章已经整理好并准备投稿。另外，我们关于mixing性质和power regular性质得到了部分结果，我们将进一步研究这些性质并将努力给出完整刻画。此外，我们考虑了几乎反交换自伴矩阵一致逼近问题，得到了反交换版本的Davidson扩张定理。我们将继续考虑其他逼近问题，这将有助于理解Hilbert空间算子的性质和结构。