Composition and nuclearity of kernel operators

Let T and S be kernel operators of finite double norm, respectively finite inverse double norm. We prove that the compositions TS and ST are nuclear operators, provided one of the function norms is order continuous. Conversely we prove theorems about factorizations of nuclear kernel operators on Banach function spaces. These results are then used to derive optimal eigenvalue estimates for nuclear operators on Banach lattices by relating Pisier's results with the results of König and Weis.