$\ell^1$-Bounded Sets

TL;DR

This paper investigates the properties and relationships of $\,\ell^1$-bounded and frame-bounded sets in separable Hilbert spaces, exploring their structural characteristics and posing open problems with potential implications.

Contribution

It introduces the concept of $\,\ell^1$-bounded sets, analyzes their properties, operations, and connections to frame-bounded sets, and presents open problems in the area.

Findings

Properties of $\,\ell^1$-bounded sets are derived.

Operations on $\,\ell^1$-bounded sets are characterized.

Relations between $\,\ell^1$-boundedness and frame-boundedness are established.

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 $\sup_{x \in M} \sum_{n \in \mathbb{N}} |\langle x, e_n\rangle| < \infty.$ 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.