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
In this paper, we first determine the relations among the best bounds A and B of the g-frame, the g-frame operator S and the pre-frame operator Q and give a necessary and sufficient condition for a g-frame with bounds...In this paper, we first determine the relations among the best bounds A and B of the g-frame, the g-frame operator S and the pre-frame operator Q and give a necessary and sufficient condition for a g-frame with bounds A and B in a complex Hilbert space. We also introduce the definition of a g-frame sequence and obtain a necessary and sufficient condition for a g-frame sequence with bounds A and B in a complex Hilbert space. Lastly, we consider the stability of a g-frame sequence for a complex Hilbert space under perturbation.展开更多
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.
基金supported by Natural Science Foundation of Fujian Province of China (No.2009J01007)Education Commission Foundation of Fujian Province of China (No.JA08013)
文摘In this paper, we first determine the relations among the best bounds A and B of the g-frame, the g-frame operator S and the pre-frame operator Q and give a necessary and sufficient condition for a g-frame with bounds A and B in a complex Hilbert space. We also introduce the definition of a g-frame sequence and obtain a necessary and sufficient condition for a g-frame sequence with bounds A and B in a complex Hilbert space. Lastly, we consider the stability of a g-frame sequence for a complex Hilbert space under perturbation.
