Frames and outer frames for Hilbert C^*-modules

The goal of the present paper is to extend the theory of frames for countably generated Hilbert $C^{\ast }$-modules over arbitrary $C^{\ast }$-algebras. In investigating the non-unital case we introduce the concept of outer frame as a sequence in the multiplier module $M(X)$ that has the standard frame property when applied to elements of the ambient module $X$. Given a Hilbert $\text{A}$-module $X$, we prove that there is a bijective correspondence of the set of all adjointable surjections from the generalized Hilbert space ${\mathrm{\ell }}^2(\text{A})$ to $X$ and the set consisting of all both frames and outer frames for $X$. Building on a unified approach to frames and outer frames we then obtain new results on dual frames, frame perturbations, tight approximations of frames and finite extensions of Bessel sequences.