On Dual $K$-$g$-Bessel Sequences and $K$-$g$-Orthonormal Bases

Abstract

In Hilbert spaces, $K$-$g$-frames are an advanced version of $g$-frames that enable the reconstruction of objects from the range of a bounded linear operator $K.$ This research investigates $K$-$g$-frames in Hilbert space. Firstly, using the $g$-preframe operators, we characterize the dual $K$-$g$-Bessel sequence of a $K$-$g$ frame. We provide additional requirements that must be met for the sum of a given $K$-$g$-frame and its dual $K$-$g$-Bessel sequence to be a $K$-$g$-frame. At the end of this paper, we present the concept of $K$-$g$-orthonormal bases and explain their link to $g$-orthonormal bases in Hilbert space. We also provide an alternative definition of $K$-$g$-Riesz bases using $K$-$g$-orthonormal bases. This gives a better understanding of the concept.