Continuity of Linear Operators in Projective and Inductive Limits of Sequences of Quasi-Banach Spaces of Holomorphic Function

We establish some conditions under which some classical operators acting between countable projective (inductive) limits of weighted quasi-Banach spaces of functions holomorphic in a plain domain are continuous. It is obtained abstract criteria for the continuity of a linear operator on projective (inductive) limits of an arbitrary sequence of quasi-Banach spaces of holomorphic functions which are stated in terms of delta-functions. The above results are applied to the weighted composition operator (including multiplication and usual composition ones). As a consequence, we obtain criteria of the continuity of the above-mentioned operators on certain weighted spaces with integral norms. Namely, we state some criteria for the continuity of weighted composition operators acting from projective (inductive) limits of weighted Bergman, Hardy, Dirichlet spaces and derivative Hardy spaces.