Continuous $p$-Bessel mappings and continuous $p$-frames in Banach spaces

Abstract

We define the concept of continuous $p$-frames ($cp$-frames) for Banach spaces, generalizing discrete $p$-frames. We prove that under certain conditions the direct sum of a finite number of $cp$-frames is again a $cp$-frame. We obtain equivalent conditions for duals of $cp$-Bessel mappings and show existence and uniqueness of duals of independent $cp$-frames. Lastly we discuss perturbation of these frames.