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Extreme points in quotients of Hardy spaces

Konstantin M. Dyakonov

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.

Extreme points in quotients of Hardy spaces

Abstract

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

Paper Structure

This paper contains 5 sections, 5 theorems, 53 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:hone']}
  4. Proof of Theorem \ref{['thm:kertoe']}
  5. Proof of Theorem \ref{['thm:hinfty']}

Key Result

Theorem 1.1

Let $E=\theta H^1_\Phi$, with $\theta$ an inner function and $\Phi$ a subset of $C({\mathbb T})$. If $\theta$ is constant, assume in addition that $\Phi$ contains the constant function $1$. Given a unit-norm coset $f+E$ in $H^1/E$, the following are equivalent: (i.1) $f+E$ is an extreme point of $\t

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof