Conditional quasi-greedy bases in Hilbert and Banach spaces

TL;DR

This paper establishes bounds on the growth of conditionality constants for quasi-greedy bases in Hilbert spaces, proves their optimality, and constructs examples with large constants in Banach spaces, advancing understanding of basis behavior.

Contribution

It provides the first non-trivial bounds on conditionality constants for quasi-greedy bases in Hilbert spaces and constructs explicit examples demonstrating their optimality.

Findings

Conditionality constants grow at most as O((log N)^{1-ε}) in Hilbert spaces.

The bounds on growth are proven to be optimal through explicit constructions.

Examples of quasi-greedy bases with large conditionality constants are constructed in Banach spaces.

Abstract

We show that, for quasi-greedy bases in Hilbert spaces, the associated conditionality constants grow at most as $O(\log N)^{1-\epsilon}$, for some $\epsilon>0$, answering a question by Temlyakov. We show the optimality of this bound with an explicit construction, based on a refinement of the method of Olevskii. This construction leads to other examples of quasi-greedy bases with large $k_N$ in Banach spaces, which are of independent interest.