[Paper Review] Weighted composition operators of $C_0(X)$'s

Jyh-Shyang Jeang, Ngai‐Ching Wong|ArXiv.org|Jul 30, 1996

Holomorphic and Operator TheoryMathematics19 citations

TL;DR

This paper establishes that into isometries and disjointness-preserving linear maps from $C_0(X)$ into $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces, are essentially weighted composition operators of the form $Tf = h \cdot f \circ \varphi$. The key result shows that such maps arise from a continuous, proper map $\varphi$ and a bounded, non-vanishing weight function $h$, with $|h| \equiv 1$ in the isometric case and $h \neq 0$ in the disjointness-preserving case.

ABSTRACT

In this paper, we prove that into isometries and disjointness preserving linear maps from $C_0(X)$ into $C_0(Y)$ are essentially weighted composition operators $Tf = h\cdot f\circφ$ for some continuous map $φ$ and some continuous scalar-valued function $h$.

Motivation & Objective

• To characterize the structure of linear maps from $C_0(X)$ into $C_0(Y)$ that are either into isometries or disjointness-preserving.
• To extend classical Banach-Stone-type theorems to non-surjective and non-compact settings by identifying weighted composition operators as the canonical form.
• To demonstrate that the standard approach of extending to one-point compactifications fails in general, necessitating new proofs for locally compact spaces.
• To provide a complete characterization of bounded disjointness-preserving linear maps on $C_0(X)$ as weighted composition operators.
• To resolve the limitations of prior compact space results by constructing counterexamples showing that extensions to compactifications may not preserve map type.

Proposed method

• Use of the cozero set and support of functions to analyze the behavior of linear maps and their interaction with function supports.
• Definition of the support of a functional $\delta_y \circ T$ to locate the image of points under the associated map $\varphi$.
• Application of properness of maps $\varphi$ to ensure that $f \circ \varphi \in C_0(Y)$ when $f \in C_0(X)$, crucial for mapping into $C_0(Y)$.
• Construction of a counterexample in Section 4 to show that extending an into isometry or disjointness-preserving map to the one-point compactification does not preserve the map type.
• Use of approximation via sequences $f_n$ with shrinking supports to derive contradictions in the behavior of $T_\infty$ at infinity.
• Employment of measure-theoretic interpretation of $\delta_y \circ T$ as a Borel measure to analyze point evaluations and derive continuity and boundedness constraints.

Experimental results

Research questions

• RQ1 Can every into isometry from $C_0(X)$ into $C_0(Y)$ be represented as a weighted composition operator?
• RQ2 Under what conditions is a bounded disjointness-preserving linear map from $C_0(X)$ into $C_0(Y)$ necessarily a weighted composition operator?
• RQ3 Why do standard extension techniques from compact to locally compact spaces fail for preserving isometry or disjointness properties?
• RQ4 What structural constraints must the map $\varphi$ and weight $h$ satisfy for $Tf = h \cdot f \circ \varphi$ to map $C_0(X)$ into $C_0(Y)$?
• RQ5 Can the behavior of $T$ at infinity be characterized via measures associated with point evaluations $\delta_y \circ T$?

Key findings

• Every into isometry $T: C_0(X) \to C_0(Y)$ is equivalent to a weighted composition operator $Tf = h \cdot f \circ \varphi$ on a locally compact subset $Y_1 \subset Y$, where $\varphi: Y_1 \to X$ is a proper continuous surjection and $|h(y)| \equiv 1$ for all $y \in Y_1$.
• Every bounded disjointness-preserving linear map $T: C_0(X) \to C_0(Y)$ is equivalent to a weighted composition operator $Tf = h \cdot f \circ \varphi$ on an open subset $Y_1 \subset Y$, where $\varphi: Y_1 \to X$ is continuous and $h(y) \neq 0$ for all $y \in Y_1$.
• The counterexample in Section 4 shows that extending an into isometry or disjointness-preserving map $T: C_0(X) \to C_0(Y)$ to the one-point compactifications $X_\infty$ and $Y_\infty$ does not preserve the map type, invalidating a naive extension strategy.
• The functional $g(y) = m_y(\{\infty\})$, representing the mass at infinity, must satisfy $g(y)g(-y) = 0$ for $|y| > 2$ in the isometric case, leading to a contradiction unless $|g(y)| \leq 1$, which fails under the assumption of an isometric extension.
• The continuity of $T_\infty \mathbf{1}$ on $Y_\infty$ and the requirement $\|T_\infty \mathbf{1}\| = 1$ force incompatible limits at $+\infty$ and $-\infty$ if $T_\infty$ were an isometry, contradicting $|g(y)| \leq 1$.
• If $T_\infty$ were disjointness-preserving, the identity $T_\infty f_n \cdot T_\infty (\mathbf{1} - f_{2n}) = 0$ would force $T_\infty \mathbf{1}(y) = h(y)$ on intervals where $h(y) \neq 0$, leading to $\lim_{y \to \infty} h(y) = 1$ and $\lim_{y \to -\infty} h(y) = -1$, contradicting continuity and boundedness.