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

TL;DR

This paper explores conditions under which conditional bases in quasi-Banach spaces can be renormed to become isometric, extending classical results to more general, non-Schauder bases using geometric and operator techniques.

Contribution

It introduces new methods to renorm quasi-Banach spaces with spreading and symmetric bases, generalizing isometric renorming theorems beyond unconditional and Schauder bases.

Findings

Spreading bases are automatically seminormalized and uniformly spreading.

Symmetric bases are necessarily spreading and uniformly symmetric.

Spaces with symmetric bases can be renormed to make all permutations isometries.

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.