FPT Lecture 3d - a case study part 1: non-unitary isometries

Following Chapter 2 of the course book, this lecture focuses on a case study involving non-unitary isometries. We will compute the distribution of this example, which corresponds to the Catalan numbers, and examine its analytic distribution that is highly significant—a semi-circular element. To achieve this, we will introduce the Cauchy transform, a crucial tool in free probability (and beyond).