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

Jaikishan, Sneh Lata, Dinesh Singh

TL;DR

The paper generalizes GKZ and KS-type rigidity results to vector-valued and multivariable settings by proving a KS-type decomposition for topological spaces of $\mathcal{A}$-valued functions, then applying this to Hardy spaces to obtain weighted composition operator representations and point-evaluation structure. It also establishes a multiplicative GKZ analogue in Hardy spaces, showing that spectrum- or image-constrained multiplicative maps must be point-evaluations, and it discusses implications for disc-algebra mappings and operator theory on Hardy-type spaces. Collectively, the results extend GKZ/KS-type characterizations beyond Banach algebras to broader, multivariate, and vector-valued contexts, with concrete operator-theoretic applications.

Abstract

In this paper, we provide a version of the classical result of Kowalski and Słodkowski that generalizes the famous Gleason-Kahane-$\dot{\rm Z}$elazko (GKZ) theorem by characterizing multiplicative linear functionals amongst all complex-valued functions on a Banach algebra. We first characterize maps on $\mathcal{A}$-valued polynomials of several variables that satisfy some conditions, motivated by the result of Kowalski and Słodkowski, as a composition of a multiplicative linear functional on $\mathcal{A}$ and a point evaluation on the polynomials, where $\mathcal{A}$ is a complex Banach algebra with identity. We then apply it to prove an analogue of Kowalski and Słodkowski's result on topological spaces of vector-valued functions of several variables. These results extend our previous work from \cite{jaikishan2024multiplicativity}; however, the techniques used differ from those used in \cite{jaikishan2024multiplicativity}. Furthermore, we characterize weighted composition operators between Hardy spaces over the polydisc amongst the continuous functions between them. Additionally, we register a partial but noteworthy success toward a multiplicative GKZ theorem for Hardy spaces.

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

TL;DR

The paper generalizes GKZ and KS-type rigidity results to vector-valued and multivariable settings by proving a KS-type decomposition for topological spaces of -valued functions, then applying this to Hardy spaces to obtain weighted composition operator representations and point-evaluation structure. It also establishes a multiplicative GKZ analogue in Hardy spaces, showing that spectrum- or image-constrained multiplicative maps must be point-evaluations, and it discusses implications for disc-algebra mappings and operator theory on Hardy-type spaces. Collectively, the results extend GKZ/KS-type characterizations beyond Banach algebras to broader, multivariate, and vector-valued contexts, with concrete operator-theoretic applications.

Abstract

In this paper, we provide a version of the classical result of Kowalski and Słodkowski that generalizes the famous Gleason-Kahane-elazko (GKZ) theorem by characterizing multiplicative linear functionals amongst all complex-valued functions on a Banach algebra. We first characterize maps on -valued polynomials of several variables that satisfy some conditions, motivated by the result of Kowalski and Słodkowski, as a composition of a multiplicative linear functional on and a point evaluation on the polynomials, where is a complex Banach algebra with identity. We then apply it to prove an analogue of Kowalski and Słodkowski's result on topological spaces of vector-valued functions of several variables. These results extend our previous work from \cite{jaikishan2024multiplicativity}; however, the techniques used differ from those used in \cite{jaikishan2024multiplicativity}. Furthermore, we characterize weighted composition operators between Hardy spaces over the polydisc amongst the continuous functions between them. Additionally, we register a partial but noteworthy success toward a multiplicative GKZ theorem for Hardy spaces.
Paper Structure (4 sections, 15 theorems, 28 equations)

This paper contains 4 sections, 15 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Kowalski-Sł odkowski's Theorem for Hardy spaces
  3. Kowalski-Sł odkowski's theorem for topological spaces
  4. A Multiplicative GKZ Theorem for Hardy spaces

Key Result

Theorem 1.1

Let $F$ be a linear functional on a complex unital Banach algebra $\mathcal{A}$ with identity $e$ such that $F(e)=1$ and $F(x)\ne0$ for every $x\in G(\mathcal{A})$, then $F(xy)=F(x)F(y)$ for all $x,y\in\mathcal{A}$.

Theorems & Definitions (25)

  • Theorem 1.1: GKZ theorem
  • Theorem 1.2: KS theorem
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • ...and 15 more