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Topologizability and Power Boundedness of Convolutions and Toeplitz Operators on Power Series Spaces

Nazlı Doğan

TL;DR

The work extends Toeplitz-operator theory to non-normable Fréchet spaces by analyzing when convolution, dual convolution, and Toeplitz operators act topologizably and with power boundedness on power series spaces $\Lambda_{1}(\alpha)$ and $\Lambda_{\infty}(\alpha)$. It derives necessary and sufficient criteria for the topologizability, m-topologizability, and power boundedness of $\widehat{T}_{\theta}$ and $\widecheck{T}_{\beta}$, expressed via growth of convolution powers $\theta^{*k}$ and $\beta^{*k}$ in the relevant Köthe norms, with stable or nuclear conditions ensuring well-definedness. For Toeplitz operators $T_{\theta,\beta}$ on nuclear spaces $\Lambda_{1}(n)$ and $\Lambda_{\infty}(n)$, strong tameness yields m-topologizability and, under small-norm bounds, power boundedness; the results transfer to holomorphic function spaces $H(\mathbb{C})$ and $H(\mathbb{D})$ via established isomorphisms. Collectively, the paper provides concrete criteria for ergodic and spectral-type properties of Toeplitz-type operators in non-normable settings and broadens the scope of operator-theoretic techniques in function-analytic contexts.

Abstract

We characterize the topologizability and power boundedness of convolution and dual convolution operators on power series spaces. We determine necessary conditions for a Toeplitz operator to be m-topologizable, and power bounded on $Λ_{1}(n)$ and $Λ_{\infty}(n)$, and consequently on $H(\mathbb{C})$ and $H(\mathbb{D})$.

Topologizability and Power Boundedness of Convolutions and Toeplitz Operators on Power Series Spaces

TL;DR

The work extends Toeplitz-operator theory to non-normable Fréchet spaces by analyzing when convolution, dual convolution, and Toeplitz operators act topologizably and with power boundedness on power series spaces and . It derives necessary and sufficient criteria for the topologizability, m-topologizability, and power boundedness of and , expressed via growth of convolution powers and in the relevant Köthe norms, with stable or nuclear conditions ensuring well-definedness. For Toeplitz operators on nuclear spaces and , strong tameness yields m-topologizability and, under small-norm bounds, power boundedness; the results transfer to holomorphic function spaces and via established isomorphisms. Collectively, the paper provides concrete criteria for ergodic and spectral-type properties of Toeplitz-type operators in non-normable settings and broadens the scope of operator-theoretic techniques in function-analytic contexts.

Abstract

We characterize the topologizability and power boundedness of convolution and dual convolution operators on power series spaces. We determine necessary conditions for a Toeplitz operator to be m-topologizable, and power bounded on and , and consequently on and .

Paper Structure

This paper contains 10 sections, 39 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Toeplitz Operators on Power Series Spaces
  4. The Convolution Operators $\widehat{T}_{\theta}$ Acting on Power Series Spaces
  5. The Dual Convolution Operators $\widecheck{T}_{\beta}$ Acting on Power Series Spaces
  6. Products Within Convolution and Dual Convolution Operator Classes
  7. Topologizability and Power Boundedness of Convolution Operators on Power Series Spaces
  8. Topologizability and Power Boundedness of Dual Convolution Operators on Power Series Spaces
  9. Topologizability and Power Boundedness of Toeplitz Operators $\mathbf{T_{\theta,\beta}}$ on $\mathbf{\Lambda_{1}(n)}$ and $\mathbf{\Lambda_{\infty}(n)}$
  10. Acknowledgements

Key Result

Lemma 2.1

Let $K(a_{n,k})$ and $K(b_{n,k})$ be Köthe spaces.

Theorems & Definitions (56)

  • Lemma 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8: Theorem 2.5,K1
  • Theorem 3.1
  • ...and 46 more