Компактность линейных операторов на квазибанаховых пространствах голоморфных функций

Abstract

We state conditions under which some classical operators acting from abstract quasi-Banach spaces of functions holomorphic in a plain domain into a weighted space of the same functions with sup-norm are compact. It is obtained abstract criteria for the compactness of a linear operator on an arbitrary quasi-Banach space which are stated in terms of delta-functions and formulate their realizations for both classical and generalized Fock spaces. The above results are applied to the weighted composition operator. It is established some conditions for the compactness of this operator which are given in terms of norms of delta-functions in the corresponding dual spaces. These results are essential generalizations of the known Zorboska’s ones. Namely, we significantly extended the class of weighted spaces of holomorphic functions with uniform norms for which one can state some conditions for the compactness of an arbitrary linear operator or the weighted composition operator.