Perturbation by weakly continuous forms and semigroups on Hardy space

In the first part of the article perturbation of a closed form by a weakly continuous form is studied. This notion of weakly continuous perturbation is very handy and, as is shown in the article, leads to a new semigroup whose difference with the given semigroup consists of compact operators. We apply the results to elliptic operators on the Hardy space generalizing results from Semigroup Forum 95(2017), 281-292. A holomorphic semigroup operating on the Hardy space is obtained whose asymptotic behaviour is studied and which is compared with the semigroup generated by the elliptic operator with periodic boundary conditions on L2(0,2π).