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Multiplication Operator Semigroups on Banach lattice valued continuous function spaces

Tobi David Olabiyi

TL;DR

This work characterizes when multiplication operators on the Banach-lattice valued function space $C_0(\Omega,E)$ generate $C_0$-semigroups. Under the assumption that the pointwise semigroups are uniformly continuous, the authors prove that there exists a continuous function $\phi: \Omega \to \mathcal{Z}(E)_s$ such that the semigroup acts pointwise as $\mathcal{T}_\phi(t)s(x)=e^{t\phi(x)}s(x)$ and is generated by the (generally unbounded) multiplication operator $\mathcal{M}_\phi$. The semigroup is uniformly continuous iff $\phi$ is bounded, i.e., $\phi \in C_b(\Omega, \mathcal{Z}(E)_s)$. The paper also provides a complete spectral description of $\mathcal{M}_\phi$, discusses the maximal domain, and includes illustrative examples with both bounded and unbounded generators, extending known scalar results to Banach-lattice valued spaces. These results have implications for multiplier algebras, center structures of Banach lattices, and the analysis of operator semigroups on vector-valued function spaces.

Abstract

We introduce and characterize, on the Banach lattice valued continuous function space, multiplication operators generating strongly continuous multiplication operator semigroups. Our characterization is the generalization of known results for the scalar-valued continuous functions $C_0(Ω)$ vanishing at infinity, on a locally compact (Hausdorff) space $Ω$, to Banach lattice $C_0(Ω, E)$ of continuous Banach lattice $E$-valued functions vanishing at infinity.

Multiplication Operator Semigroups on Banach lattice valued continuous function spaces

TL;DR

This work characterizes when multiplication operators on the Banach-lattice valued function space generate -semigroups. Under the assumption that the pointwise semigroups are uniformly continuous, the authors prove that there exists a continuous function such that the semigroup acts pointwise as and is generated by the (generally unbounded) multiplication operator . The semigroup is uniformly continuous iff is bounded, i.e., . The paper also provides a complete spectral description of , discusses the maximal domain, and includes illustrative examples with both bounded and unbounded generators, extending known scalar results to Banach-lattice valued spaces. These results have implications for multiplier algebras, center structures of Banach lattices, and the analysis of operator semigroups on vector-valued function spaces.

Abstract

We introduce and characterize, on the Banach lattice valued continuous function space, multiplication operators generating strongly continuous multiplication operator semigroups. Our characterization is the generalization of known results for the scalar-valued continuous functions vanishing at infinity, on a locally compact (Hausdorff) space , to Banach lattice of continuous Banach lattice -valued functions vanishing at infinity.

Paper Structure

This paper contains 6 sections, 4 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Multiplication Operators on $C_0(\Omega, E)$.
  3. Multiplication semigroup on $C_0(\Omega, E)$ generated by $(\mathcal{M}_\phi, D(\mathcal{M}_\phi) )$
  4. Characterization of multiplication semigroups on $C_0(\Omega, E)$
  5. Proof of Theorem \ref{['thm1']}:
  6. Examples

Key Result

Theorem 1

Let $m_t\in C_b(\Omega, \boldsymbol{ \mathcal{Z} }(E)_\textbf{s})$ for each $t\ge0$ such that the induced multiplication semigroup $\mathcal{T}(t)_{t\ge0}$ given by is a $C_0$-semigroup on $C_0(\Omega, E)$. And denote its (infinitesimal) generator on $C_0(\Omega, E)$ by $(\mathcal{M}, D(\mathcal{M}))$. Furthermore, suppose that, for each $x\in\Omega$, the induced pointwise multiplication semigro

Theorems & Definitions (10)

  • Theorem 1
  • Definition 1
  • Proposition 2
  • proof
  • Remark 3
  • Definition 4
  • Proposition 5
  • proof
  • Lemma 1
  • proof