Generalized frames for operators associated with atomic systems

Abstract

In this paper, we investigate the g-frame and Bessel g-sequence related to a linear bounded operator $K$ in Hilbert space, which we call a $K$-g-frame and a $K$-dual Bessel g-sequence, respectively. Since the frame operator for a $K$-g-frame may not be invertible, there is no classical canonical dual for a $K$-g-frame. So we characterize the concept of a canonical $K$-dual Bessel g-sequence of a $K$-g-frame that generalizes the classical dual of a g-frame. Moreover, we use a family of linear operators to characterize atomic systems. We also consider the construction of new atomic systems from given ones and bounded operators.