Spectral Properties for the Magnetization Integral Operator

We analyze the spectrum of a certain singular integral operator on the space (L²(Ω ))³ where Ω is contained in three dimensional Euclidean space and has a Lipschitz continuous boundary. This operator arises in the integral formulation of the magnetostatic field problem. We decompose (L²(Ω ))³ into invariant subspaces: in one where the operator is the zero map; in one, the identity map; and in one where the operator is positive definite and bounded. These results give rise to the formulation of new efficient numerical techniques for approximating nonlinear magnetostatic field problems [5], [6], [12].