The spectral theorem for quaternionic unbounded normal operators based\n on the S-spectrum

In this paper we prove the spectral theorem for quaternionic unbounded normal\noperators using the notion of $S$-spectrum. The proof technique consists of\nfirst establishing a spectral theorem for quaternionic bounded normal operators\nand then using a transformation which maps a quaternionic unbounded normal\noperator to a quaternionic bounded normal operator. With this paper we complete\nthe foundation of spectral analysis of quaternionic operators. The $S$-spectrum\nhas been introduced to define the quaternionic functional calculus but it turns\nout to be the correct object also for the spectral theorem for quaternionic\nnormal operators. The fact that the correct notion of spectrum for quaternionic\noperators was not previously known has been one of the main obstructions to\nfully understanding the spectral theorem in this setting. A prime motivation\nfor studying the spectral theorem for quaternionic unbounded normal operators\nis given by the subclass of unbounded anti-self adjoint quaternionic operators\nwhich play a crucial role in the quaternionic quantum mechanics.\n