“无界圆盘”上$H^{\infty }$商空间的范数可达性

The Norm-Attaining Property of the $H^{\infty }$ Quotient Space over the “Unbounded Disk”

摘要: 交换子提升是算子理论的核心问题，由Sarason率先提出。其经典工作成功实现了单位圆周上Hardy空间$H^2$ 的子空间——模型空间上保持范数的交换子提升。借鉴Sarason的方法框架，证明“无界圆盘”边界${\mathbb{T}}_0$ 上交换子提升的范数保持性，需要验证$H^{\infty }$ 空间的如下性质：商空间$H^{\infty }\left({\mathbb{T}}_0\right)/{\psi }_0{H}^{\infty }\left({\mathbb{T}}_0\right)$ 的每个陪集，均包含一个在$H^{\infty }\left({\mathbb{T}}_0\right)$ 中范数达到陪集范数的函数。本文首先介绍“无界圆盘”上Hardy空间的基础知识，随后给出该性质的完整证明。

Abstract: Commutant lifting is a core problem in operator theory, first proposed by Sarason. His classic work successfully realized the norm-preserving commutant lifting on model spaces, which are subspaces of the Hardy space $H^2$ over the unit circle. Drawing on Sarason’s framework, to prove the norm-preserving property of commutant lifting on the boundary ${\mathbb{T}}_0$ of the “unbounded disk”, the following property of the $H^{\infty }$ space needs to be verified: every coset in the quotient space $H^{\infty }\left({\mathbb{T}}_0\right)/{\psi }_0{H}^{\infty }\left({\mathbb{T}}_0\right)$ contains a function whose norm in $H^{\infty }\left({\mathbb{T}}_0\right)$ achieves the coset norm. This paper first introduces the preliminaries of Hardy spaces over the “unbounded disk”, and then presents a complete proof of this property.