Spectrum and analytic functional calculus in real and quaternionic frameworks: An overview

An approach to the elementary spectral theory for quaternionic linear operators was presented by the author in a recent paper, quoted and discussed in the Introduction, where, unlike in works by other authors, the construction of the analytic functional calculus used a Riesz-Dunford-Gelfand type kernel, and the spectra were defined in the complex plane. In fact, the present author regards the quaternionic linear operators as a special class of real linear operators, a point of view leading to a simpler and a more natural approach to them. The author's main results in this framework are summarized in the following, and other pertinent comments and remarks are also included in this text. In addition, a quaternionic joint spectrum for pairs of operators is discussed, and an analytic functional calculus which uses a Martinelli type kernel in two variables is recalled.