Spectral Properties of the Reflection Operator in Two Dimensions
We study spectral properties of the reflection operator (a singular integral operator arising naturally in connection with the radiosity equation which models the energy transfer between different parts of a surface by radiation) acting on L p spaces, p ∈ (1, ∞), on infinite angles in two dimensions. More specifically we establish an explicit characterization of the spectrum and spectral radius estimates for the reflection operator acting on L p spaces on an infinite angle in two dimensions. This type of analysis is relevant to the solvability of the radiosity equation with L p data since when the spectral radius is < 1, the solution can be explicitly expressed as a convergent Neumann series.
I. Mitrea, K. Ott, E. Stachura, Spectral Properties of the Reflection Operator in Two Dimensions. Contemporary Mathematics, Vol. 581, pp. 199-215, 2012.