Kernels of adjoints of composition operators with rational symbols of degree two

摘要

In 2006, Cowen and Gallardo-Gutiérrez laid the groundwork for an explicit formula for the adjoint of a composition operator with rational symbol acting on the Hardy space, and in 2008, Hammond, Moorhouse, and Robbins established such a formula. While it is known that the kernel of the adjoint of a composition operator whose symbol is not univalent on the complex unit disk is infinite-dimensional, no classification has been given for functions in this kernel. Using the adjoint formula, we give a functional equation that characterizes functions in the kernel of the adjoint of a composition operator with a rational symbol of degree two.