Weak $A$-frames and weak $A$-semi-frames

TL;DR

This paper introduces the concept of weak lower A-semi-frames, exploring their properties, duality, and examples, expanding the theoretical framework of frames controlled by operators.

Contribution

It defines and analyzes weak lower A-semi-frames, a new class of frames controlled by operators, and compares them with existing lower atomic systems.

Findings

Introduction of weak lower A-semi-frames

Comparison with lower atomic systems

Discussion of duality and examples

Abstract

After reviewing the interplay between frames and lower semi-frames, we introduce the notion of lower semi-frame controlled by a densely defined operator $A$ or, for short, a \emph{weak lower $A$-semi-frame} and we study its properties. In particular, we compare it with that of lower atomic systems, introduced in \cite{GB}. We discuss duality properties and we suggest several possible definitions for weak $A$-upper semi-frames. Concrete examples are presented.