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Optimal upper and lower sequence spaces with applications

Sergey V. Astashkin, Per G. Nilsson

Abstract

We study the optimal upper $X_U$ and lower $X_L$ sequence spaces that can be assigned to each Banach lattice $X$. These spaces are symmetric, have the Fatou property and the unit vector basis has in these spaces very special properties. Determined by the order structure of $X$ the spaces $X_U$ and $X_L$ turn out to be very useful when studying Banach lattices. Among other results, in terms of these constructions, we identify Banach lattices that satisfy equal-norm upper and lower $p$-estimates, give a characterization of $L_p(μ)$-spaces, derive some properties of the tensor product operator in Lorentz and Orlicz spaces, identify Orlicz spaces in which the unit vector basis is upper (resp. lower) semi-homogeneous.

Optimal upper and lower sequence spaces with applications

Abstract

We study the optimal upper and lower sequence spaces that can be assigned to each Banach lattice . These spaces are symmetric, have the Fatou property and the unit vector basis has in these spaces very special properties. Determined by the order structure of the spaces and turn out to be very useful when studying Banach lattices. Among other results, in terms of these constructions, we identify Banach lattices that satisfy equal-norm upper and lower -estimates, give a characterization of -spaces, derive some properties of the tensor product operator in Lorentz and Orlicz spaces, identify Orlicz spaces in which the unit vector basis is upper (resp. lower) semi-homogeneous.
Paper Structure (16 sections, 20 theorems, 115 equations)

This paper contains 16 sections, 20 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries.
  3. Banach lattices, upper and lower estimates for disjoint elements.
  4. Banach function spaces
  5. Symmetric sequence spaces
  6. Lower/upper semi-homogeneous bases in Banach spaces
  7. Optimal Upper and Lower Sequence Lattices
  8. Definitions and general properties
  9. Duality type properties of optimal sequence spaces
  10. Optimal sequence spaces and order estimates of Banach lattices
  11. A characterization of $L_p(\mu)$-spaces in terms of optimal sequence spaces.
  12. A characterization of optimal sequence spaces.
  13. Tensor product in semi-homogeneous symmetric spaces
  14. Applications to Lorentz and Orlicz sequence spaces
  15. Lorentz spaces
  16. ...and 1 more sections

Key Result

Theorem 3.1

For every Banach lattice $X$, the spaces $X_{U}$ and $X_{L}$ are symmetric.

Theorems & Definitions (32)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • Example 3.7
  • ...and 22 more