Limits of kernel operators and the spectral regularity lemma

We study the spectral aspects of the graph limit theory. We give a description of graphon convergence in terms of convergence of eigenvalues and eigenspaces. Along these lines we prove a spectral version of Szemerédi’s regularity lemma. Using spectral methods we investigate group actions on graphons. As an application we show that the set of isometry invariant graphons on the sphere is closed in terms of graph convergence, however the analogous statement does not hold for the circle. This fact is rooted in the representation theory of the orthogonal group.