G-frames with bounded linear operators

Abstract

In this paper, we introduce the more general g-frame which is called a $K$-g-frame by combining a g-frame with a bounded linear operator $K$ in a Hilbert space. We give several equivalent characterizations for $K$-g-frames and discuss the stability of perturbation for $K$-g-frames. We also investigate the relationship between a $K$-g-frame and the range of the bounded linear operator $K$. In the end, we give two sufficient conditions for the remainder of a $K$-g-frame after an erasure to still be a $K$-g-frame. It turns out that although $K$-g-frames share some properties similar to g-frames, a large part of $K$-g-frames behaves completely different from g-frames.