Slice hyperholomorphicity of the $S$-resolvent operators and boundary conditions

The foundation of spectral theory on the $S$-spectrum can be traced back to the quaternionic framework of quantum mechanics. The concept of $S$-spectrum for quaternionic operators emerged as the natural spectrum in slice hyperholomorphic functional calculi, known as the $S$-functional calculus and also utilized in the quaternionic spectral theorem. This spectral theory extends to Clifford operators. A key distinction from classical complex spectral theory lies in the definition of the $S$-spectrum, which is second order in the operator $T$, and in the $S$-resolvent operators that turns out to be the product of two different operators. This study delves into the analyticity of the $S$-resolvent operators under specified boundary conditions for the $S$-spectral problem. The spectral theory on the $S$-spectrum also provides deeper insights into classical spectral theory.