Multivariable Schur–Horn theorems
Multivariable Schur–Horn theorems
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多变量 Schur–Horn 定理
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Mohan Ravichandran
中科院分区：
文献类型：
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作者：
Pedro Massey;Mohan Ravichandran
我们证明了描述 II 类 1 因子中可交换厄米算子元组对角线的各种结果。这些结果是由 Arveson 和 Kadison 的工作推动的，是经典 Schur-Horn 定理到无限维、多变量设置的推广。我们对这些可能的对角线的描述使用了经典主化概念的自然概括。在特殊情况下，当给定的元组和所需的对角线都具有有限的联合谱时，我们的结果是完整的。当元组没有有限联合谱时，我们能够证明强近似结果。与单变量情况不同，多变量情况带来了一些惊喜，我们指出了将有限谱情况中的完整描述扩展到一般情况的障碍。我们还讨论了 B(H) 中交换元组对角线的表征问题，并给出了这种情况下的近似表征。
We prove a variety of results describing the diagonals of tuples of commuting hermitian operators in type II 1 factors. These results, motivated by work of Arveson and Kadison, are generalizations of the classical Schur‐Horn theorem to the infinite‐dimensional, multivariable setting. Our description of these possible diagonals uses a natural generalization of the classical notion of majorization. In the special case when both the given tuple and the desired diagonal have finite joint spectrum, our results are complete. When the tuples do not have finite joint spectrum, we are able to prove strong approximate results. Unlike the single variable case, the multivariable case presents several surprises and we point out obstructions to extending our complete description in the finite spectrum case to the general case. We also discuss the problem of characterizing diagonals of commuting tuples in B(H) and give approximate characterizations in this case as well.