Extreme points in quotients of Hardy spaces

TL;DR

This paper characterizes the extreme points of unit balls in quotient Hardy spaces, extending classical results from $H^1$ and $H^e$ to these quotient spaces under specific conditions.

Contribution

It provides new characterizations of extreme points in quotient Hardy spaces, including cases involving kernels of Toeplitz operators.

Findings

Characterizations of extreme points in $H^1/E$ and $H^e/E$

Extension of classical theorems to quotient spaces

Analysis of the kernel of Toeplitz operators in this context

Abstract

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