Continuous slice functional calculus in quaternionic Hilbert spaces

TL;DR

This paper develops a continuous functional calculus for bounded normal operators in quaternionic Hilbert spaces using slice quaternionic functions, extending spectral theory and functional analysis in quaternionic settings.

Contribution

It introduces a new continuous functional calculus based on slice functions for quaternionic normal operators, generalizing spectral theory in quaternionic Hilbert spaces.

Findings

Defined a continuous functional calculus for bounded normal operators

Established spectral map theorems in the quaternionic setting

Extended some results to unbounded operators

Abstract

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic $C^*$--algebras and to define, on each of these $C^*$--algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.