Spectral theorem for quaternionic compact normal operators

Abstract

In this article, we prove two versions of the spectral theorem for quaternionic compact normal operators, namely the series representation and the resolution of identity form. Though the series representation form already appeared in [5], we prove this by using simultaneous diagonalization. Whereas the resolution of identity is new in the literature for the quaternion case, we prove this by associating a complex linear operator to the given right linear operator and applying the classical result. In this process we prove some spectral properties of compact operators parallel to the classical theory. We also establish the singular value decomposition of a compact operator.