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Essential norms of pointwise multipliers in the non-algebraic setting

Tomasz Kiwerski, Jakub Tomaszewski

Abstract

Motivated by some recent results, but also referring to recognized classics, we compute the essential norm and the weak essential norm of multiplication operators acting between two distinct K{\" o}the spaces both defined over the same $σ$-finite measure space. A by-product of the technology we have developed here are some applications to Banach sequence spaces related to decreasing functions and to Banach spaces of analytic functions on the unit disc, in particular, Hardy spaces. We will close our work with some specific examples illustrating the previously obtained results including Musielak--Orlicz sequence spaces (in particular, Nakano sequence spaces) as well as Lorentz and Marcinkiewicz sequence spaces.

Essential norms of pointwise multipliers in the non-algebraic setting

Abstract

Motivated by some recent results, but also referring to recognized classics, we compute the essential norm and the weak essential norm of multiplication operators acting between two distinct K{\" o}the spaces both defined over the same -finite measure space. A by-product of the technology we have developed here are some applications to Banach sequence spaces related to decreasing functions and to Banach spaces of analytic functions on the unit disc, in particular, Hardy spaces. We will close our work with some specific examples illustrating the previously obtained results including Musielak--Orlicz sequence spaces (in particular, Nakano sequence spaces) as well as Lorentz and Marcinkiewicz sequence spaces.
Paper Structure (39 sections, 25 theorems, 258 equations)

This paper contains 39 sections, 25 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Motivation and goals
  3. Neverending story
  4. Outline
  5. Statements and declarations
  6. Acknowledgments
  7. Funding
  8. Conflict of interest
  9. Data availibility
  10. Toolbox
  11. Recollections on function spaces
  12. Boyd indices
  13. Essential norms
  14. Pointwise multipliers
  15. Pointwise products
  16. ...and 24 more sections

Key Result

Lemma 3.23

Let $(\Omega,\Sigma,\mu)$ be a non-atomic complete and $\sigma$-finite measure space. Then for any function, say $f$, from $L_1(\Omega,\Sigma,\mu)$ we have

Theorems & Definitions (53)

  • Remark 3.8: Köthe function and sequence spaces
  • Remark 3.12
  • Remark 3.14
  • Example 3.16
  • Remark 3.17
  • Example 3.19
  • Remark 3.20
  • Lemma 3.23
  • Theorem 4.1: Essential norm of multiplication operators between Köthe function spaces
  • Corollary 4.2
  • ...and 43 more