The $S$-resolvent estimates for the Dirac operator on hyperbolic and spherical spaces

Abstract: This seminal paper marks the beginning of our investigation into 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 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 a very explicit form. This fact is very important for the application 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; 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. 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.