Sums of Hermitian Operators and Connections to Connes' Embedding Problem; Hyperinvariant Subspaces

本文将研究II_1因子中包含的算子理论中的两个基本问题。第一个是Connes的嵌入问题。Collins和Dykema最近的工作表明，这个问题等价于有限von Neumann代数中算子和的问题，而Bercovici、Collins、Dykema、Li和Timotin最近的工作也积极地回答了这个问题的第一部分，表明所有Horn不等式在所有有限von Neumann代数中都成立。第二个问题是超不变子空间问题。特别地，PI将关注II_1-factors元素的剩余开放部分，即II_1-factors中拟幂零算子的情况。无限维希尔伯特空间上的算子用于量子力学的数学模型，在数学的各个领域都有重要意义。我们将研究算子理论中的两个基本问题：cones嵌入问题和超不变子空间问题。它们涉及无穷维希尔伯特空间上算子结构的不同方面。我们将集中讨论其代数具有迹的算子。第一个问题是有限维空间上的算子如何很好地近似这些算子（在它们相对于轨迹的混合矩中）。我们将通过检验算子和的特征值来解决这个问题。第二个问题是关于在无限维空间上通过将算子限制在不变子空间来分解算子的可能性。在最近的一些进展中，Haagerup和Schultz已经证明了一类算子的子空间的存在性，我们将重点讨论一些这个问题尚未解决的特定算子。

AbstractDykemaThe PI will investigate two fundamental problems in the theory of operators that are contained in II_1 factors. The first is Connes' embedding problem. Recent work of Collins and Dykema has shown that this problem is equivalent to a question about sums of operators in finite von Neumann algebras, and other recent work of Bercovici, Collins, Dykema, Li and Timotin has positively answered the first part of this question, showing that all Horn inequalities hold in all finite von Neumann algebras. The second problem is the hyperinvariant subspace problem. In particular, the PI will focus on the remaining open part of this problem for elements of II_1-factors, namely, the case of quasi-nilpotent operators in II_1-factors.Operators on infinite dimensional Hilbert space are used in mathematical models of quantum mechanics, and they are of significance in diverse areas of mathematics. We will work on two fundamental problems in operator theory: Connes' embedding problem and the hyperinvariant subspace problem. These concern different aspects of the structure of operators on infinite dimensional Hilbert spaces. We will focus on operators whose algebras possess traces. The first problem is about how well such operators can be approximated (in their mixed moments with respect to the trace) by operators on finite dimensional spaces. We will attack this problem by examining eigenvalues of sums of operators. The second problem is about the possibility of decomposing operators on infinite dimensional space by restricting them to invariant subspaces. In some recent progress, Haagerup and Schultz have proved the existence of such subspaces for a large class of operators, and we will focus on some specific operators for which this question is unresolved.