关于无限矩阵、Schur积和算子测度
On infinite matrices, Schur products, and operator measures
摘要：
在量子力学中，数值在希尔伯特空间的子空间上的半双线性形式集合中的测度是很有意义的，因为它们可以被解释为只有有限的一组可能的测量准备的可观测量。在本文中，我们考虑的问题，在什么条件下，这样的措施扩展到一个运营商值的措施，在具体的设置，其中的措施是定义在Borel集的区间$[0,2π)$和关于移位的协变。在这种情况下，测度被刻画为一个单一的无限矩阵，证明了可扩性的一个基本充分条件是矩阵是Schur乘子。相应地，我们还研究了可拓性问题与Schur乘子理论之间的联系。特别地，我们定义了Schur乘子的一些新的范数。
英文摘要：
Measures with values in the set of sesquilinear forms on a subspace of a Hilbert space are of interest in quantum mechanics, since they can be interpreted as observables with only a restricted set of possible measurement preparations. In this paper, we consider the question under which conditions such a measure extends to an operator valued measure, in the concrete setting where the measure is defined on the Borel sets of the interval $[0,2π)$ and is covariant with respect to shifts. In this case, the measure is characterized with a single infinite matrix, and it turns out that a basic sufficient condition for the extensibility is that the matrix be a Schur multiplier. Accordingly, we also study the connection between the extensibility problem and the theory of Schur multipliers. In particular, we define some new norms for Schur multipliers.
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