Abstract
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.
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This research is partly supported by the Natural Science Foundation of Fujian Province, China (No. Z0511013) and the Education Commission Foundation of Fujian Province, China (No. JB04038)
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Zhu, Y.C. Characterizations of g-frames and g-Riesz bases in Hilbert spaces. Acta. Math. Sin.-English Ser. 24, 1727–1736 (2008). https://doi.org/10.1007/s10114-008-6627-0
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DOI: https://doi.org/10.1007/s10114-008-6627-0