On the exposed points of the unit ball of $H^1$

A point of a convex set is said to be exposed if there exists a real functional that achieves its maximum only at this point. Any exposed point is extreme, but the converse is not true. The problem of describing the exposed points of the unit ball of the Hardy class $H^1$ remains open. In the talk it will be shown that an outer function from $H^1$ is exposed if its absolute values are not very small (in a certain implicit sense) at each point of the circle, that is, the failure of this property means that the function is sufficiently small at least at one point. The proof is based on a result about the existence of spectral projections for almost unitary operators on a Hilbert space.