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Centered weighted composition operators on $L^2$-spaces revisited

Piotr Budzyński

Abstract

Centered weighted composition operators on $L^2$-spaces are characterized. The characterization is obtained without the assumption that the operator is a product of a multiplication and a composition operator. The concept of spectrally half-centered operators is introduced, and it is shown that unbounded weighted composition operators are spectrally half-centered provided their powers are closed and densely defined. A criteria for centered weighted shifts on directed trees of types I--IV are provided. Various examples are presented.

Centered weighted composition operators on $L^2$-spaces revisited

Abstract

Centered weighted composition operators on -spaces are characterized. The characterization is obtained without the assumption that the operator is a product of a multiplication and a composition operator. The concept of spectrally half-centered operators is introduced, and it is shown that unbounded weighted composition operators are spectrally half-centered provided their powers are closed and densely defined. A criteria for centered weighted shifts on directed trees of types I--IV are provided. Various examples are presented.
Paper Structure (7 sections, 15 theorems, 83 equations, 3 figures)

This paper contains 7 sections, 15 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General operator theory
  4. Weighted composition operators
  5. Main results
  6. Examples
  7. Acknowledgments

Key Result

Lemma 1

Let $n\in\mathbb N$, $k\in\mathbb Z_+$. Assume that $C_{\phi, w}$ is well defined and $C_{\phi^n, {w}_n}$ is densely defined. Then the following is satisfied In particular, if $C_{\phi, w}\in\boldsymbol B(L^2(\mu))$, then the following conditions hold:

Figures (3)

  • Figure 1: The directed tree ${\mathscr T}$ considered in Example \ref{['blackblack+']}.
  • Figure 2: The directed tree ${\mathscr T}$ considered in Example \ref{['exa:Y_tree_not_type1']}.
  • Figure 3: The directed tree ${\mathscr T}$ isomorphic to $\mathbb Z_-$, considered in Example \ref{['exa:Z_minus_type2']}.

Theorems & Definitions (42)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Remark 3
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • ...and 32 more