  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.
