Continuity of the linearized forward map of electrical impedance tomography from square-integrable perturbations to Hilbert-Schmidt operators

This work considers the Fréchet derivative of the idealized forward map of two-dimensional electrical impedance tomography, i.e., the linear operator that maps a perturbation of the coefficient in the conductivity equation over a bounded two-dimensional domain to the linear approximation of the corresponding change in the Neumann-to-Dirichlet boundary map. It is proved that the Fréchet derivative is bounded from the space of square-integrable conductivity perturbations to the space of Hilbert–Schmidt operators on the mean-free $L^2$ functions on the domain boundary, if the background conductivity coefficient is constant and the considered simply-connected domain has a $C^{1,\alpha }$ boundary. This result provides a theoretical framework for analyzing linearization-based one-step reconstruction algorithms of electrical impedance tomography in an infinite-dimensional setting.