Composition operators on reproducing kernel Hilbert spaces

Abstract: This paper investigates composition and weighted composition operators acting between Reproducing Kernel Hilbert Spaces (RKHS). We employ the properties of their reproducing kernel structure to provide a characterization of the boundedness of these operators, which naturally subsumes many results on boundedness of composition operators in the literature. Emphasis is put on Hardy and Bergman spaces on the unit ball $\mathbb{B}_n$ and unit polydisc $\mathbb{D}^n$. Further, we provide an alternative proof for the boundedness of composition operators on the Hardy space of the unit disc, based on reproducing kernel techniques which is different from traditional analytic and operator-theoretic methods.