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Schauder-Orlicz decompositions, $\ell_Φ$-decompositions and pseudo-Daugavet property

Vitalii Marchenko

Abstract

The concept of $\ell_Φ$-decomposition, extending the concept of $\ell_{p}$-decomposition of a Banach space, is presented and basic properties of Schauder-Orlicz decompositions and $\ell_Φ$-decompositions are studied. We show that Schauder-Orlicz decompositions are orthogonal in a sense of Grinblyum-James and Singer. Simple constructions of $\ell_{p}$-decompositions and Schauder-Orlicz decompositions in $L_p$ are presented. We prove that in the class of spaces possessing pseudo-Daugavet property, which includes classical $L_p$, $1\leq p\neq 2$, and $C$, Schauder-Orlicz decompositions with at least one finite dimensional subspace do not exist. It follows that Kato theorem on similarity for sequences of projections [1] cannot be extended to spaces from this class. Moreover we show that Banach spaces, possessing Schauder-Orlicz decompositions with at least one finite dimensional subspace, do not have pseudo-Daugavet property. Thus for Banach spaces $X$ possessing Schauder-Orlicz decompositions we obtain the following characterization of pseudo-Daugavet property: $X$ has pseudo-Daugavet property if and only if there is no Schauder-Orlicz decomposition in $X$ with at least one finite dimensional subspace if and only if there is no Schauder-Orlicz decomposition in $X$, which is an FDD.

Schauder-Orlicz decompositions, $\ell_Φ$-decompositions and pseudo-Daugavet property

Abstract

The concept of -decomposition, extending the concept of -decomposition of a Banach space, is presented and basic properties of Schauder-Orlicz decompositions and -decompositions are studied. We show that Schauder-Orlicz decompositions are orthogonal in a sense of Grinblyum-James and Singer. Simple constructions of -decompositions and Schauder-Orlicz decompositions in are presented. We prove that in the class of spaces possessing pseudo-Daugavet property, which includes classical , , and , Schauder-Orlicz decompositions with at least one finite dimensional subspace do not exist. It follows that Kato theorem on similarity for sequences of projections [1] cannot be extended to spaces from this class. Moreover we show that Banach spaces, possessing Schauder-Orlicz decompositions with at least one finite dimensional subspace, do not have pseudo-Daugavet property. Thus for Banach spaces possessing Schauder-Orlicz decompositions we obtain the following characterization of pseudo-Daugavet property: has pseudo-Daugavet property if and only if there is no Schauder-Orlicz decomposition in with at least one finite dimensional subspace if and only if there is no Schauder-Orlicz decomposition in , which is an FDD.
Paper Structure (10 sections, 29 theorems, 68 equations)

This paper contains 10 sections, 29 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Schauder-Orlicz decompositions and $\ell_{\Phi}$-decompositions
  3. Basic properties of Schauder-Orlicz decompositions and $\ell_{\Phi}$-decompositions
  4. Inclinations of subspaces, Grinblyum-James orthogonality and Schauder-Orlicz decompositions
  5. To the question of uniqueness of $\ell_{\Phi}$-decompositions and Schauder-Orlicz decompositions
  6. Existence of Schauder-Orlicz decompositions and $\ell_p$-decompositions in spaces $L_p$, $1\leq p<\infty$
  7. Nonexistence of Schauder-Orlicz decompositions with at least one finite dimensional subspace
  8. Nonexistence of Schauder-Orlicz decompositions with at least one finite dimensional subspace in spaces $L_p$, $1\leq p\neq 2$ and $C$
  9. Nonexistence of Schauder-Orlicz decompositions with at least one finite dimensional subspace in Banach spaces with pseudo-Daugavet property
  10. Concluding remarks and perspectives

Key Result

Theorem 3

Let $\{\mathfrak{M}_n\}_{n=0}^{\infty}$ be a Schauder decomposition of $X$. (i) If $\{\mathfrak{M}_n\}_{n=0}^{\infty}$ is $\ell_{\Phi}$-decomposition with non-degenerate Orlicz function $\Phi$, then $\{\mathfrak{M}_n\}_{n=0}^{\infty}$ is boundedly complete. (ii) If $\{\mathfrak{M}_n\}_{n=0}^{\infty}

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Theorem 3
  • proof
  • Corollary 4
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Corollary 7
  • ...and 41 more