Explore the second part of a mini-course on Operators on Function Spaces presented by William Ross from the University of Richmond. Delve into advanced topics such as multiplication operators, bilateral shifts, Toeplitz operators, and their matrix representations. Examine the Brown-Halmos theorem, compactness, and analytical Toeplitz operators. Investigate the commutant of the unilateral shift, spectral properties, and kernels of Toeplitz operators. Gain insights into nearly invariant subspaces and conclude with an inspirational result from Hilbert. This 51-minute lecture, part of the Focus Program on Analytic Function Spaces and their Applications, offers a deep dive into complex mathematical concepts for those interested in function space theory and operator algebra.