The spectral theorem for quaternionic unbounded normal operators based on the \(S\)-spectrum (Q2795583)

A kind of a spectral theorem for quaternionic bounded normal operators was known since the paper by \textit{K. Viswanath} [Trans. Am. Math. Soc. 162, 337--350 (1972; Zbl 0234.47024)]. However, in this and some subsequent papers, the role of some notion of spectrum, similar to the one known for operators over \(\mathbb C\), was not elucidated. The case of unbounded operators has not been considered.NEWLINENEWLINEIn the paper under review, these problems are solved. The authors use the recent notion of \(S\)-spectrum and the continuous functional calculus; see [the second author et al., Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions. Progress in Mathematics 289. Basel: Birkhäuser (2011; Zbl 1228.47001); \textit{R. Ghiloni} et al., Rev. Math. Phys. 25, No. 4, Article ID 1350006, 83 p. (2013; Zbl 1291.47008)]. The case of unitary operators was considered by the authors earlier [Milan J. Math. 84, No. 1, 41--61 (2016; Zbl 1342.35189)].