Hilbert-Schauder Frame Operators

TL;DR

This paper introduces Hilbert-Schauder frame operators, blending Hilbert space frame theory with Banach space Schauder frames, providing new structural insights and characterizations of Hilbert spaces.

Contribution

It defines a new class of frame operators for Banach spaces, bridging Hilbert and Banach frame theories, and offers a novel isomorphic characterization of Hilbert spaces.

Findings

Examples include standard Hilbert frames and bases of l_p and L^p spaces for 1< p   2

Basic structural properties of the Hilbert-Schauder frame operator are established

Provides a new isomorphic characterization of Hilbert spaces

Abstract

We introduce a new concept of frame operators for Banach spaces we call a Hilbert-Schauder frame operator. This is a hybird between standard frame theory for Hilbert spaces and Schauder frame theory for Banach spaces. Most of our results involve basic structure properties of the Hilbert-Schauder frame operator. Examples of Hilbert-Schauder frames include standard Hilbert frames and classical bases of $\ell_p$ and $L^p$-spaces with $1< p \le 2$. Finally, we give a new isomorphic characterization of Hilbert spaces.