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 $A$.
Categories: math.FA