Slice Hyperholomorphicity of the S-resolvent Operators and Boundary Conditions

Abstract 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.