Theory of Integral Operators in Parametric Clifford Type Algebras

Abstract

We develop a theory of integral operators in the Clifford type associative algebra depending on parameters, generated by the structure relations: $$\begin{aligned} e_ie_j+e_je_i = -2 B_{ij}, \end{aligned}$$for $$i,j=1,2,\ldots ,n$$, where the $$B_{ij} $$ are the entries of a symmetric and positive definite matrix $$B \in \mathbb {R}^{n \times n}$$. As applications we consider some Dirichlet boundary problems for the Dirac operator $${\mathcal {D}}$$ in this algebra and the elliptic operator $$\tilde{\Delta }_n = div(B\nabla )$$.