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

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 weak lower $A$-semi-frame and we study its properties. In particular, we compare it with that of lower atomic systems, introduced in (GB). We discuss duality properties and we suggest several possible definitions for weak $A$-upper semi-frames. Concrete examples are presented.