Conditional quasi-greedy bases in Hilbert and Banach spaces

Abstract

For quasi-greedy bases B in Hilbert spaces, we give an improved bound of the associated conditionality constants kN (B) = O(logN)1−e, for some e > 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 kN in Banach spaces, which are of independent interest.