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Remarks about symmetry-type conditions of conditional bases of Banach spaces

José L. Ansorena, Alejandro Marcos

TL;DR

The paper addresses when conditional bases in quasi-Banach spaces admit isometric renormings that render symmetry operations into isometries. It develops a spreading-model framework to obtain an equivalent $p$-norm for spreading bases, making all increasing maps isometries, and separately leverages uniform symmetry for symmetric bases to achieve permutation isometries via a semigroup renorming lemma. These results unify and extend classical isometric renorming theorems from unconditional, subsymmetric, and symmetric Schauder bases to the conditional, non-Schauder setting, using spreading models and operator semigroups as central tools. The findings have implications for the geometric structure of quasi-Banach spaces and the stability of their bases under symmetry operations, providing a cohesive renorming theory for both spreading and symmetric bases.

Abstract

We investigate the existence of equivalent p-norms, 0< p 1, under which conditional symmetric or spreading bases in quasi-Banach spaces become isometric. For spreading bases (which need not be unconditional or even Schauder bases), we develop new techniques involving the geometry of spreading sequences and their associated spreading models. We prove that any spreading basis is automatically seminormalized, M-bounded, and uniformly spreading, which allows the construction of an isometric renorming via its spreading model. For symmetric bases, we show they are necessarily spreading and uniformly symmetric, enabling a direct application of a renorming lemma for uniformly bounded semigroups of operators. Consequently, any quasi-Banach space with a symmetric basis admits a renorming making all permutations isometries, and any spreading basis admits a renorming making all increasing maps isometries. These results extend and unify classical isometric renorming theorems for unconditional, subsymmetric, and symmetric Schauder bases to the conditional, non-Schauder setting.

Remarks about symmetry-type conditions of conditional bases of Banach spaces

TL;DR

The paper addresses when conditional bases in quasi-Banach spaces admit isometric renormings that render symmetry operations into isometries. It develops a spreading-model framework to obtain an equivalent -norm for spreading bases, making all increasing maps isometries, and separately leverages uniform symmetry for symmetric bases to achieve permutation isometries via a semigroup renorming lemma. These results unify and extend classical isometric renorming theorems from unconditional, subsymmetric, and symmetric Schauder bases to the conditional, non-Schauder setting, using spreading models and operator semigroups as central tools. The findings have implications for the geometric structure of quasi-Banach spaces and the stability of their bases under symmetry operations, providing a cohesive renorming theory for both spreading and symmetric bases.

Abstract

We investigate the existence of equivalent p-norms, 0< p 1, under which conditional symmetric or spreading bases in quasi-Banach spaces become isometric. For spreading bases (which need not be unconditional or even Schauder bases), we develop new techniques involving the geometry of spreading sequences and their associated spreading models. We prove that any spreading basis is automatically seminormalized, M-bounded, and uniformly spreading, which allows the construction of an isometric renorming via its spreading model. For symmetric bases, we show they are necessarily spreading and uniformly symmetric, enabling a direct application of a renorming lemma for uniformly bounded semigroups of operators. Consequently, any quasi-Banach space with a symmetric basis admits a renorming making all permutations isometries, and any spreading basis admits a renorming making all increasing maps isometries. These results extend and unify classical isometric renorming theorems for unconditional, subsymmetric, and symmetric Schauder bases to the conditional, non-Schauder setting.
Paper Structure (3 sections, 17 theorems, 71 equations)

This paper contains 3 sections, 17 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Renorming quasi-Banach spaces with spreading bases
  3. Renorming quasi-Banach spaces with symmetric bases

Key Result

Lemma 1.1

Let $0<p\le 1$ and ${\mathbb{X}}$ be a locally $p$-convex quasi-Banach space. Suppose that ${\mathcal{G}}$ is a uniformly bounded semigroup of endomorphisms of ${\mathbb{X}}$. Then, there is an equivalent $p$-norm for ${\mathbb{X}}$ relative to which $\left\lVert T\right\rVert\le 1$ for all $T\in {

Theorems & Definitions (40)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 30 more