Every semi-normalized unconditional Schauder frame in Hilbert spaces contains a frame

Abstract

Let $H$ be an infinite-dimensional Hilbert space. We prove that every unconditional Schauder frame for $H$ contains a subsequence that can be normalized to form a frame for $H$. As a consequence, every semi-normalized unconditional Schauder frame contains a frame for $H.$ Here we say that a sequence $\{x_n\}_{n\in \mathbb{N}}$ in a Hilbert space $H$ is an \emph{unconditional Schauder frame} for $H$ if there exists some sequence $\{y_n\}_{n\in \mathbb{R}}subseteq H$ such that $$x=\sum_{n=1}^\infty \langle x,y_n\rangle x_n\quad \text{for all }x\in H,$$ with the unconditional convergence of the series in the norm of $H.$ We say that $\{x_n\}_{n\in\mathbb{N}}$ is semi-normalized if $m\leq \|x_n\|\leq M$ for all $n\in \mathbb{N}$ for some positive constants $m,M.$ We then apply our main results to answer several open questions concerning the existence of certain unconditional Schauderf frames. For example, we prove that if a closed subspace of $L^2(\mathbb{R}^d)$ contains $\{e^{2πib\cdot x}g\}_{b\in Λ}$ for some infinite uniformly discrete subset $Λ$ of $\mathbb{R}^d$ and some nonzero function $g$ in the Feichtinger algebra, it does not admit any unconditional Schauder frames of translates with finitely many generators. We will also show that no Gabor system with the critical lower Beurling density can be an unconditional Schauder frame when the window function belongs to the Feichtinger algebra. Furthermore, we present an example of a compact set of $\mathbb{R}$ which does not admit any unconditional Schauder frames of exponentials with the critical lower Beurling density. All results in this paper apply equivalently to sequences that can be rescaled to form a frame for $H.$