#### Title

### Spectral Properties of the Reflection Operator in Two Dimensions

#### Document Type

Journal Article

#### Role

Author

#### Standard Number

0271-4132

#### Journal Title

Contemporary Mathematics

#### Volume

581

#### First Page

199

#### Last Page

215

#### Publication Date

2012

#### Abstract

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.

#### Repository Citation

I. Mitrea, K. Ott, E. Stachura, Spectral Properties of the Reflection Operator in Two Dimensions. Contemporary Mathematics, Vol. 581, pp. 199-215, 2012.