${\ell }^1$-Bounded Sets

Abstract

A subset $M$ of a separable Hilbert space $H$ is ${\ell }^1$-bounded if there exists a Riesz basis $\mathcal{F}=\{e_n{\}}_{n\in \mathbb{N}}$ for $H$ such that ${\sup }_{x\in M}{\sum }_{n\in \mathbb{N}}|⟨x,e_n⟩|<\infty .$ A similar definition for ${\ell }^1$-frame-bounded sets is made by replacing Riesz bases with frames. This paper derives properties of ${\ell }^1$-bounded sets, operations on the collection of ${\ell }^1$-bounded sets, and the relation between ${\ell }^1$-boundedness and ${\ell }^1$-frame-boundedness. Some open problems are stated, several of which have intriguing implications.