Frames and weak frames for unbounded operators

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