Frames and outer frames for Hilbert -modules

Abstract

The goal of the present paper is to extend the theory of frames for countably generated Hilbert -modules over arbitrary -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 -module X, we prove that there is a bijective correspondence of the set of all adjointable surjections from the generalized Hilbert space 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.