An introduction to model spaces and their operators

This work presents a comprehensive study of asymmetric truncated Toeplitz operators (ATTOs), which are operators acting between two different model spaces, defined by inner functions. Building upon foundational results for classical and truncated Toeplitz operators, we investigate the algebraic properties and products of these operators.The main results provide necessary and sufficient conditions for the product of two ATTOs to itself be an asymmetric truncated Toeplitz operator.