A spectral theory for direct integrals of operators
A spectral theory for direct integrals of operators
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算子直接积分的谱理论
DOI：
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发表时间：
1970
期刊：
影响因子：
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通讯作者：
T. R. Chow
中科院分区：
文献类型：
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作者：
T. R. Chow
N. Dunford [4] 于 1966 年建立了希尔伯特空间有限直和上某些算子的谱理论。在本文中，我们将使用冯·诺依曼的约简理论方法（参见 [2, 7]）证明他的定理 2.7, 1-4] 对于更广泛的算子类别是正确的。事实上，我们证明了希尔伯特空间中的任何闭谱算子都可以分解为闭不可约谱算子的直积分。在第 1 节和第 3 节中，我们将包括一些关于有界和无界算子约简理论的已经开发的结果。我们的主要结果在第二节和第四节。最后一节包括将闭谱算子分解为不可约部分的方法，其中我们还指出了我们将来要研究的结构理论问题。我们的成果的各种应用将出现在其他地方。在本文中，希尔伯特空间将位于复数域上。作者感谢 N. Suzuki 教授对这个问题的建议以及与他进行的许多有价值的讨论。
A spectral theory for certain operators on a finite direct sum of Hilbert spaces was established in 1966 by N. Dunford [4]. In this paper, we shall show by using the approach of reduction theory of von Neumann (cf. [2, 7]) that his Theorem 2.7, 1-4] is true for a wider class of operators. In fact, we proved that any closed psectral operator in a Hilbert space can be decomposed into a direct integral of closed irreducible spectral operators. In Sections 1 and 3, we shall include some already developed results about the reduction theory of bounded and unbounded operators. Our main results are in Sections 2 and 4. A method of decomposition of closed spectral operator into irreducible part is included in the last section, in which we also indicate a problem of structure theory that we shall investigate in the future. The various applications of our results will appear elsewhere. Throughout this paper, the Hilbert space will be over the complex field. The author is grateful to Professor N. Suzuki for his suggestion of the problem and many valuable sessions of discussion with him.