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A Pełczyński-Vogt decomposition result for (PLS)-spaces and sequence space representations

Andreas Debrouwere, Lenny Neyt

TL;DR

This work extends the classical Pelczynski–Vogt decomposition to (PLS)-spaces, proving a splitting result and a PV decomposition for infinite-type power-series spaces $\Lambda_{\infty}(\alpha,\beta)$. By coupling these results with Gabor frame techniques, it derives sequence-space representations for multiplier spaces of Beurling-type Gelfand–Shilov spaces, yielding explicit identifications like $\mathcal{Z}^{(\mu)}_{(\tau)} \cong \Lambda_{\infty}(\alpha_{\mu},\alpha_{\tau})$ for suitable parameters. The approach hinges on a robust splitting framework for $(\text{PLS})$-spaces and on time-frequency tools to translate operator-topological properties into concrete sequence-space descriptions. The findings provide a unified pathway to represent and analyze multiplier spaces in ultradifferentiable and ultradistri-bution settings, with potential applications in harmonic analysis and distribution theory.

Abstract

We establish a Pelczyński-Vogt decomposition result for (PLS)-type power series spaces of infinite type. By combining this result with the theory of Gabor frames, we obtain sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type.

A Pełczyński-Vogt decomposition result for (PLS)-spaces and sequence space representations

TL;DR

This work extends the classical Pelczynski–Vogt decomposition to (PLS)-spaces, proving a splitting result and a PV decomposition for infinite-type power-series spaces . By coupling these results with Gabor frame techniques, it derives sequence-space representations for multiplier spaces of Beurling-type Gelfand–Shilov spaces, yielding explicit identifications like for suitable parameters. The approach hinges on a robust splitting framework for -spaces and on time-frequency tools to translate operator-topological properties into concrete sequence-space descriptions. The findings provide a unified pathway to represent and analyze multiplier spaces in ultradifferentiable and ultradistri-bution settings, with potential applications in harmonic analysis and distribution theory.

Abstract

We establish a Pelczyński-Vogt decomposition result for (PLS)-type power series spaces of infinite type. By combining this result with the theory of Gabor frames, we obtain sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type.

Paper Structure

This paper contains 22 sections, 24 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. A Peł czyński-Vogt decomposition result for (PLS)-spaces
  3. Sequence space representations for multiplier spaces of Gelfand-Shilov spaces of Beurling type
  4. Outline of the paper
  5. Preliminaries
  6. Notation
  7. Projective spectra
  8. $(\mathop{\mathrm{PLS}}\nolimits)$-spaces
  9. Short exact sequences
  10. Sequence spaces
  11. Fréchet- and $(\mathop{\mathrm{LB}}\nolimits)$-type sequence spaces
  12. $(\mathop{\mathrm{PLB}}\nolimits)$-type sequence spaces
  13. A splitting result for $(\mathop{\mathrm{PLS}}\nolimits)$-type sequence spaces
  14. Locally splitting spectra
  15. A sufficient condition for the vanishing of $\mathop{\mathrm{Proj}}\nolimits^{1} L(E, X)$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\mu, \tau > 1/2$. Set $\alpha_\mu = (j^{1/\mu})_{j \in \mathbb{N}}$ and $\alpha_\tau = (j^{1/\tau})_{j \in \mathbb{N}}$. Then,

Theorems & Definitions (51)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 41 more