Hilbert空间中K-框架的若干问题研究

K-框架是在研究有界线性算子的原子分解时引入的, 它是经典框架的推广, 即K-框架用新的下界取代了经典框架的下框架条件.本项目着重研究Hilbert空间中K-框架的若干问题:(1)引入K-Riesz框架的Gram矩阵和冗余度概念, 并借助Gram矩阵研究K-Riesz框架与K-Riesz基的性质及其等价刻画, 进而研究K-Riesz框架在扰动下的稳定性, 并探讨K-Riesz框架冗余度;(2)基于K-g-框架在实际应用中较g-框架有一定的优势, 在解决与K-g-框架相关的两个g-Bessel序列之间的交换性问题基础上, 探讨K-g-框架与紧K-g-框架的性质、构造方法和稳定性;(3)在深入分析框架、g-框架的等式和不等式基础上, 给出K-框架和K-g-框架的等式与不等式;(4)刻画控制K-框架与控制K-g-框架性质, 寻找有效的控制K-框架与控制K-g-框架构造方法, 并研究它们的稳定性.

K-frames have been introduced in studying the atomic systems with respect to a bounded linear operator as a generalization of classical frames, i.e. K-frames replace the lower frame bound condition by new lower condition. This project focuses on several problems of K-frames for Hilbert spaces. (1) We introduce the concepts of Gram matrix and excess of K-Riesz frames, and dicuss the properties and characterizations of K-Riesz frames and K-Riesz bases by means of Gram matrix. Furthermore, by employing the enquivalent characterizations of K-Riesz frames, new results on the stability of K-Riesz frames are discussed, and the excess of K-Riesz frames is investigated. (2) K-g-frames have a certain advantage in practical applications relative to g-frames. After the exchangeable issue of two g-Bessel sequences related to K-g-frame is solved, we study the properties, construction methods and the stability of K-g-frames and tight K-g-frames. (3) Based on the elaborate analysis of identities and inequalities of frames and g-frames, we give the identities and inequalities of K-frames and K-g-frames. (4)Characterize the properties of controlled K-frames and controlled K-g-frames, seek the effective construction methods of controlled K-frames and controlled K-g-frames, and then study the stability of controlled K-frames and controlled K-g-frames.