Multiplication operators on spaces of continuous functions

DefinitionThe multiplication operator on continuous functions#✓fully proved

Let \(X\) be a compact topological space, let \(\mathbb{K}\) be \(\mathbb{R}\) or \(\mathbb{C}\), and let \(C(X, \mathbb{K})\) be the Banach algebra of continuous functions \(X \to \mathbb{K}\) with the supremum norm. For \(\varphi \in C(X, \mathbb{K})\), the multiplication operator \(M_\varphi\) is the bounded linear operator on \(C(X, \mathbb{K})\) given by \(M_\varphi f = \varphi f\).

noncomputable def mulCLM (φ : C(X, 𝕜)) : C(X, 𝕜) →L[𝕜] C(X, 𝕜) :=
  ContinuousLinearMap.mul 𝕜 C(X, 𝕜) φ

Used by 4

Used byEigenvalues of a multiplication operator · Invertibility of a multiplication operator · Multiplication by a constant on continuous functions · Spectrum of a multiplication operator

TheoremMultiplication by a constant on continuous functions#✓fully proved

Let \(X\) be a compact topological space, let \(\mathbb{K}\) be \(\mathbb{R}\) or \(\mathbb{C}\), and let \(\mu \in \mathbb{K}\). The multiplication operator on \(C(X, \mathbb{K})\) by the constant function with value \(\mu\) equals \(\mu\) times the identity operator.

theorem mulCLM_algebraMap (μ : 𝕜) :
    mulCLM (algebraMap 𝕜 C(X, 𝕜) μ) = algebraMap 𝕜 (C(X, 𝕜) →L[𝕜] C(X, 𝕜)) μ

Uses 1 · Used by 1

Statement usesThe multiplication operator on continuous functions

Used bySpectrum of a multiplication operator

TheoremSpectrum of a multiplication operator#✓fully proved

Let \(X\) be a compact topological space, let \(\mathbb{K}\) be \(\mathbb{R}\) or \(\mathbb{C}\), and let \(\varphi \in C(X, \mathbb{K})\). The spectrum of the multiplication operator \(M_\varphi f = \varphi f\), as an element of the algebra of bounded linear operators on \(C(X, \mathbb{K})\), is the range \(\varphi(X)\) of \(\varphi\).

theorem spectrum_mulCLM (φ : C(X, 𝕜)) : spectrum 𝕜 (mulCLM φ) = Set.range φ

Uses 2

Statement usesThe multiplication operator on continuous functions

Proof usesMultiplication by a constant on continuous functions

TheoremInvertibility of a multiplication operator#✓fully proved

Let \(X\) be a compact topological space, let \(\mathbb{K}\) be \(\mathbb{R}\) or \(\mathbb{C}\), and let \(\varphi \in C(X, \mathbb{K})\). The multiplication operator \(M_\varphi f = \varphi f\) is invertible in the algebra of bounded linear operators on \(C(X, \mathbb{K})\) if and only if \(\varphi(x) \neq 0\) for every \(x \in X\).

theorem isUnit_mulCLM_iff (φ : C(X, 𝕜)) :
    IsUnit (mulCLM φ) ↔ ∀ x : X, φ x ≠ 0

Uses 1

Statement usesThe multiplication operator on continuous functions

TheoremEigenvalues of a multiplication operator#✓fully proved

Let \(X\) be a compact Hausdorff space, let \(\mathbb{K}\) be \(\mathbb{R}\) or \(\mathbb{C}\), let \(\varphi \in C(X, \mathbb{K})\) and let \(\mu \in \mathbb{K}\). Then \(\mu\) is an eigenvalue of the multiplication operator \(M_\varphi f = \varphi f\) on \(C(X, \mathbb{K})\) if and only if the level set \(\varphi^{-1}(\{\mu\})\) has nonempty interior.

theorem hasEigenvalue_mulCLM_iff [T2Space X] (φ : C(X, 𝕜)) (μ : 𝕜) :
    Module.End.HasEigenvalue (mulCLM φ : C(X, 𝕜) →ₗ[𝕜] C(X, 𝕜)) μ ↔
      (interior (φ ⁻¹' {μ})).Nonempty

Uses 1

Statement usesThe multiplication operator on continuous functions

5 fully proved

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