Remark: Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane (by left multiplication). If we pick a single complex number of unit length (which we think of as "U"), it is intuitive that its powers will fill up the circle. Since the circle is symmetric around 0, it makes sense that the averages of the powers of "U" will converge to 0. Also, 0 is the only fixed point of "U", and so the projection onto the space of fixed points must be the zero operator (which agrees with the limit just described).