Extreme points in quotients of Hardy spaces

In the Hardy spaces H¹ H 1 and H^∞ H ∞ , there are neat and well-known characterizations of the extreme points of the unit ball. We obtain counterparts of these classical theorems when H¹ H 1 (resp., H^∞ H ∞ ) gets replaced by the quotient space H¹/E H 1 / E (resp., H^∞ /E H ∞ / E ), under certain assumptions on the subspace E . In the H¹ H 1 setting, we also treat the case where the underlying space is taken to be the kernel of a Toeplitz operator.