Abstract: In this talk I will introduce the Hardy space and give some historical motivation as to how one might study the Hardy space. I will then introduce a well-known operator on the Hardy space, the truncated Toeplitz operator. When Sarason initiated the study of truncated Toeplitz operators in 2007 he posed the question, does every bounded truncated Toeplitz operator have a bounded symbol? This was then shown to be negative in 2010 when a bounded truncated Toeplitz operator was constructed which does not possess a bounded symbol. Further work then gave a characterisation of when a bounded truncated Toeplitz operator has a bounded symbol. In this talk I will consider analogous questions looking at the role of continuous symbols for compact truncated Toeplitz operators.
