Dilation Theory: A Guided Tour
Dilation Theory: A Guided Tour
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膨胀理论：导览
DOI：
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发表时间：
2020
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影响因子：
--
通讯作者：
O. Shalit
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文献类型：
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作者：
O. Shalit
扩张理论是研究算子的一种范式，它通过将一个算子表现为另一个在某种意义上表现良好的算子的压缩。例如，每个收缩可以被扩张到（即，是么正算子的压缩），并在这个简单的事实上发展了一个关于非正规算子的深入理论。在第一部分的调查，我将悠闲地审查关键经典结果膨胀理论为一个单一的运营商或几个交换运营商，和样本应用膨胀理论在运营商理论和函数理论。然后，在第二部分中，我将快速介绍膨胀理论的大量变体及其应用。特别是，我将讨论膨胀理论的完全积极的地图和半群，以及算子代数方法膨胀理论。在最后一部分中，我将提出相对较新的膨胀问题，在非交换设置，这是相关的矩阵凸集和算子系统的研究，并在控制理论中的应用。这些问题包括将非交换算子的元组扩张为具有特定联合谱的交换正规算子的元组。我还将描述最近研究的确定最佳常数$c = c_{\theta，\theta '}$的问题，使得满足$VU = e^{i\theta} UV$的每对酉$U，V$可以扩张为一对$cU'，cV '$，其中$U'，V '$是满足对易关系$V' U ' = e^{i\theta'} U 'V'$的酉。这个问题的解决引起了一个新的和令人惊讶的应用膨胀理论的连续性频谱的几乎马蒂厄算子从数学物理。
Dilation theory is a paradigm for studying operators by way of exhibiting an operator as a compression of another operator which is in some sense well behaved. For example, every contraction can be dilated to (i.e., is a compression of) a unitary operator, and on this simple fact a penetrating theory of non-normal operators has been developed. In the first part of this survey, I will leisurely review key classical results on dilation theory for a single operator or for several commuting operators, and sample applications of dilation theory in operator theory and in function theory. Then, in the second part, I will give a rapid account of a plethora of variants of dilation theory and their applications. In particular, I will discuss dilation theory of completely positive maps and semigroups, as well as the operator algebraic approach to dilation theory. In the last part, I will present relatively new dilation problems in the noncommutative setting which are related to the study of matrix convex sets and operator systems, and are motivated by applications in control theory. These problems include dilating tuples of noncommuting operators to tuples of commuting normal operators with a specified joint spectrum. I will also describe the recently studied problem of determining the optimal constant $c = c_{\theta,\theta'}$, such that every pair of unitaries $U,V$ satisfying $VU = e^{i\theta} UV$ can be dilated to a pair of $cU', cV'$, where $U',V'$ are unitaries that satisfy the commutation relation $V'U' = e^{i\theta'} U'V'$. The solution of this problem gives rise to a new and surprising application of dilation theory to the continuity of the spectrum of the almost Mathieu operator from mathematical physics.
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影响因子：
0.8
作者：
Malte Gerhold;O. Shalit
通讯作者：
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作者：
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影响因子：
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作者：
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通讯作者：
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影响因子：
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作者：
M. Gerhold;O. M. Shalit
通讯作者：
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