基于S-谱的四元数无界正规算子的谱定理
The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum
摘要：
本文利用$S$ -谱的概念证明了四元数无界正规算子的谱定理。证明技术包括首先建立一个四元数有界正规算子的谱定理，然后使用一个变换，该变换将一个四元数无界正规算子映射到一个四元数有界正规算子。完成了四元数算子谱分析的基础。$S$ -谱已被引入来定义四元数谓词演算，但它也被证明是四元数正规算子谱定理的正确对象。事实上，正确的概念，频谱的四元数运营商是不知道以前的主要障碍之一，以充分理解频谱定理在这种情况下。无界反自伴四元数算子子类是研究四元数无界正规算子谱定理的一个主要动机，它在四元数量子力学中起着至关重要的作用.
英文摘要：
In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of $S$-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The $S$-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The fact that the correct notion of spectrum for quaternionic operators was not previously known has been one of the main obstructions to fully understanding the spectral theorem in this setting. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.
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