  In 2012 G\u{a}vru\c{t}a introduced the notions of $K$-frame and of atomic
system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$, in
order to decompose its range $\mathcal{R}(K)$ with a frame-like expansion. In
this article we revisit these concepts for an unbounded and densely defined
operator $A:\mathcal{D}(A)\to\mathcal{H}$ in two different ways. In one case we
consider a non-Bessel sequence where the coefficient sequence depends
continuously on $f\in\mathcal{D}(A)$ with respect to the norm of $\mathcal{H}$.
In the other case we consider a Bessel sequence and the coefficient sequence
depends continuously on $f\in\mathcal{D}(A)$ with respect to the graph norm of
$A$.
