Reproducing kernels turn questions about a composition operator C_phi(f) = f o phi, or a weighted composition operator W_{phi,psi}(f) = psi * (f o phi), into concrete identities on the kernel functions of the underlying space. Because the adjoint of such an operator acts on kernels by a simple rule, properties like adjoint form, norm attainment, and which symbols induce isometries can be read off directly rather than through Carleson-measure estimates. The technique applies across weighted Hardy spaces H^2(beta), the Hardy space of the disc, weighted Bergman spaces, and their several-variable analogues on the polydisc.
A composition operator C_phi sends a holomorphic function f on the unit disk to f composed with phi, where phi is a holomorphic self-map of the disk; the weighted version W_phi,psi(f) = psi times (f …