On Dual $K$-$g$-Bessel Sequences and $K$-$g$-Orthonormal Bases (Journal of Nonlinear Modeling and Analysis, 2024)
In Hilbert spaces, $K$-$g$-frames are an advanced version of $g$-framesthat enable the reconstruction of objects from the range of a bounded linearoperator $K.$ This research investigates $K$-$g$-frames in Hilbert space. Firstly,using the $g$-preframe operators, we characterize the dual $K$-$g$-Bessel sequenceof a $K$-$g$ frame. We provide additional requirements that must be met for thesum 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 andexplain their link to $g$-orthonormal bases in Hilbert space. We also provide analternative definition of $K$-$g$-Riesz bases using $K$-$g$-orthonormal bases. Thisgives a better understanding of the concept.
Keywords: $K$-$g$-frames; dual $K$-$g$-Bessel sequences; $K$-$g$-orthonormal bases; $K$-$g$-Riesz bases.