Characterizations of g-Frames and g-Riesz Bases in Hilbert Spaces

In this paper,we introduce the pre-frame operator Q for the g-frame in a complex Hilbertspace,which will play a key role in studying g-frames and g-Riesz bases etc.Using the pre-frameoperator Q,we give some necessary and sufficient conditions for a g-Bessel sequence,a g-frame,and ag-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 ofa g-frame or a g-Riesz basis for a Hilbert space under perturbation.