In this lecture we give an application of elementary operators-on-Hilbert-space theory, by proving von Neumann’s mean ergodic theorem. See also this treatment by Terry Tao on his blog.
For today’s lecture we will require the following simple fact which I forgot to mention in the previous one.
Exercise A: Let $latex A, B in B(H)$. Then $latex |AB| leq |A| |B|$.