【摘要】 复Hilbert空间中的K-框架是框架的一种推广,是Gǎvruta在研究算子K的原子分解系统时引入的.本文首先在Hilbert空间H中引入K-Riesz基的概念,给出H中K-Riesz基界为A和B的K-Riesz基的两个等价刻画及K-框架界为A和B的K-框架的一个特征.众所周知,H中无冗框架与Riesz基是等价的,但是无冗K-框架与K-Riesz基是不等价的.接着研究H中无冗K-框架与K-Riesz基之间的关系.最后,考虑H中K-框架或K-Riesz基的扰动的稳定性.当K为H中的恒等算子时,这些结果与框架或Riesz基的相应结果是一致的.更多还原

【Abstract】 The K-frame, which is a generalization of a frame in a complex Hilbert space H, was introduced by Gǎvruta 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.更多还原