Universality property of the S-functional calculus, noncommuting matrix variables and Clifford operators.

Spectral theory on the S -spectrum was born out of the need to give quaternionic quantum mechanics a precise mathematical foundation (Birkhoff and von Neumann [8] showed that a general set theoretic formulation of quantum mechanics can be realized on real, complex or quaternionic Hilbert spaces). Then it turned out that spectral theory on S -spectrum has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for n -tuples of noncommuting operators. With this paper we show that the spectral theory on the S -spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. In fact, the S -spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the S -functional calculus goes beyond quaternionic and Clifford analysis, indeed the S -functional calculus has a certain universality property. This fact makes the spectral theory on the S -spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case. [ABSTRACT FROM AUTHOR]