A Gleason-Kahane-Żelazko theorem for reproducing kernel Hilbert spaces

Gleason-Kahane-Żelazko theorems in function spaces

The Gleason-Kahane-Żelazko theorem states that a linear functional on a Banach algebra that is non-zero on invertible elements is necessarily a scalar multiple of a character. Recently this theorem has been extended to certain Banach function spaces that are not algebras. In this article we present a brief survey of these extensions.

A Gleason-Kahane-Żelazko theorem for modules and applications to holomorphic function spaces

We generalize the Gleason-Kahane-Żelazko theorem to modules. As an application, we show that every linear functional on a Hardy space that is non-zero on outer functions is a multiple of a point evaluation. A further consequence is that every linear endomorphism of a Hardy space that maps outer functions to nowhere-zero functions is a weighted composition operator. In neither case is continuity assumed. We also consider some extensions to other function spaces, including the Bergman, Dirichlet and Besov spaces, the little Bloch space and VMOA.

A Gleason-Kahane-Żelazko theorem for the Dirichlet space

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions. We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functio…

$H^p_{α, β}$ 空间上的 Gleason-Kahane-Żelazko 定理

A Gleason-Kahane-Żelazko Theorem for $H^p_{α, β}$ spaces

在 $H^p_{α, β}$ 空间上建立 GKZ 型定理，通过识别可逆元素的对等函数类，证明非零连续线性泛函为点赋值，并应用于刻画加权复合算子。

On the Gleason-Kahane-Żelazko theorem for associative algebras

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $Λ(\mathbf 1)=1$, is multiplicative, that is, $Λ(ab)=Λ(a)Λ(b)$ for all $a,b\in A$. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra $A$ over a field of at least $3$ elements, and having an ideal of codimension $1$, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If $A$ is a commutative algebra, then the l…

拓扑空间上的多变量Kowalski-Słodkowski定理

A several variables Kowalski-S\lodkowski theorem for topological spaces

本文将Kowalski-Słodkowski定理推广到多变量情形，通过刻画Banach代数值多项式上的映射为乘法线性泛函与点赋值的复合，进而在向量值函数拓扑空间及Hardy空间上建立相应结果。