$\boldsymbol{K}$-frames and $\boldsymbol{K}$-Riesz bases incomplex Hilbert spaces

Abstract

The K-frame, which is a generalization of a frame in a complex Hilbert space H, was introduced by Gȃvruţa to study an atomic system. In this paper, we first introduce the concept of a K-Riesz basis for H. We give a characterization of a K-frame for H with K-frame bounds A and B and two equivalent characterizations of a K-Riesz basis for H with K-Riesz basis bounds A and B. It is well known that an exact frame is equivalent to a Riesz basis in H. But an exact K-frame is not equivalent to a K-Riesz basis in H. We also study the relation between an exact K-frame and a K-Riesz basis in H. Lastly, we consider the stability of a K-frame or a K-Riesz basis for H under perturbation. All the obtained results are consistent with those of frames or Riesz bases when K is the identity operator for a complex Hilbert space.