This seminal paper marks the beginning of our investigation on the spectral theory based on S-spectrum applied to the Dirac operator on manifolds. Specifically, we examine in detail the cases of the Dirac operator \(\mathcal {D}_H\) on hyperbolic space and of the Dirac operator \(\mathcal {D}_S\) on the spherical space, where these operators, and their squares \(\mathcal {D}_H^2\) and \(\mathcal {D}_S^2\) , can be written in an explicit form. This fact is very important for the applications of the spectral theory on the S-spectrum. In fact, let T denote a (right) linear Clifford operator, the S-spectrum is associated with a second-order polynomial in the operator T, specifically the operator defined as \(Q_s(T) := T^2 - 2s_0T + |s|^2.\) This allows us to associate to the Dirac operator boundary conditions that can be of Dirichlet type but also of Robin-like type. Moreover, our theory is not limited to Hilbert modules but it is applicable to Banach modules as well. The spectral theory based on the S-spectrum has gained increasing attention in recent years, particularly as it aims to provide quaternionic quantum mechanics with a solid mathematical foundation from the perspective of spectral theory. This theory was extended to Clifford operators, and more recently, the spectral theorem has been adapted to this broader context. Moreover, the S-spectrum is crucial for defining the so-called S-functional calculus for quaternionic and Clifford operators in various forms. This includes bounded as well as unbounded operators, where suitable estimates of sectorial and bi-sectorial type for the S-resolvent operator are essential for the convergence of the Dunford integrals in this setting.
