Operator-Valued p-Approximate Schauder Frames

TL;DR

This paper develops an operator-algebraic framework for p-approximate Schauder frames, unifying various frame theories in Hilbert and Banach spaces, and extends the theory to Banach spaces with new concepts and characterizations.

Contribution

It introduces a unified operator-algebraic approach to p-approximate Schauder frames, including new Banach space concepts and characterizations.

Findings

Unified theory encompassing operator-valued frames, G-frames, and p-approximate Schauder frames.

Extension of frame concepts and properties to Banach spaces.

Development of duality, orthogonality, and stability notions in the Banach space setting.

Abstract

We give an operator-algebraic treatment of theory of p-approximate Schuader frames which includes the theory of operator-valued frames by Kaftal, Larson, and Zhang [\textit{Trans. AMS., 2009}], G-frames by Sun [JMAA, 2006], factorable weak operator-valued frames by Krishna and Johnson [\textit{Annals of FA, 2022}] and p-approximate Schauder frames by Krishna and Johnson [\textit{J. Pseudo-Differ. Oper. Appl, 2021}] as particular cases. We show that a sufficiently rich theory can be developed even for Banach spaces. We achieve this by defining various concepts and characterizations in Banach spaces. These include duality, approximate duality, equivalence, orthogonality and stability.