In this paper,we interpret the operator representation using g-frames as a generalization of U-cross Gram matrices.We establish the link between U-cross g-Gram matrices andg-Riesz bases,and obtain a characterization o...In this paper,we interpret the operator representation using g-frames as a generalization of U-cross Gram matrices.We establish the link between U-cross g-Gram matrices andg-Riesz bases,and obtain a characterization ofg-Riesz bases by U-cross g-Gram matrices.In particular,someexamples show that the invertibility of U-cross g-Gram matrix is not possible when the associated sequences are g-frames but not g-Riesz bases or at most one of them is a g-Riesz basis.Finally,we show that the invertibility of U-cross g-Gram matrices is preserved under small perturbations of the operators or the sequences.展开更多
In this paper, we introduce the pre-frame operator Q for the g-frame in a complex Hilbert space, which will play a key role in studying g-frames and g-Riesz bases etc. Using the pre-frame operator Q, we give some nece...In this paper, we introduce the pre-frame operator Q for the g-frame in a complex Hilbert space, which will play a key role in studying g-frames and g-Riesz bases etc. Using the pre-frame operator Q, we give some necessary and sufficient conditions for a g-Bessel sequence, a g-frame, and a g-Riesz basis in a complex Hilbert space, which have properties similar to those of the Bessel sequence, frame, and Riesz basis respectively. We also obtain the relation between a g-frame and a g-Riesz basis, and the relation of bounds between a g-frame and a g-Riesz basis. Lastly, we consider the stability of a g-frame or a g-Riesz basis for a Hilbert space under perturbation.展开更多
基金Supported by NSF of Henan Province (Nos.252300420353,252300421973)Key Scientific and Technological Project of Henan Province (No.242102210049)。
文摘In this paper,we interpret the operator representation using g-frames as a generalization of U-cross Gram matrices.We establish the link between U-cross g-Gram matrices andg-Riesz bases,and obtain a characterization ofg-Riesz bases by U-cross g-Gram matrices.In particular,someexamples show that the invertibility of U-cross g-Gram matrix is not possible when the associated sequences are g-frames but not g-Riesz bases or at most one of them is a g-Riesz basis.Finally,we show that the invertibility of U-cross g-Gram matrices is preserved under small perturbations of the operators or the sequences.
基金the Natural Science Foundation of Fujian Province,China (No.Z0511013)the Education Commission Foundation of Fujian Province,China (No.JB04038)
文摘In this paper, we introduce the pre-frame operator Q for the g-frame in a complex Hilbert space, which will play a key role in studying g-frames and g-Riesz bases etc. Using the pre-frame operator Q, we give some necessary and sufficient conditions for a g-Bessel sequence, a g-frame, and a g-Riesz basis in a complex Hilbert space, which have properties similar to those of the Bessel sequence, frame, and Riesz basis respectively. We also obtain the relation between a g-frame and a g-Riesz basis, and the relation of bounds between a g-frame and a g-Riesz basis. Lastly, we consider the stability of a g-frame or a g-Riesz basis for a Hilbert space under perturbation.
