【摘要】 Hilbert空间中的g-框架是框架的自然推广,它们包含了许多推广的框架,如子空间框架或fusion框架、斜框架和拟框架等.它们有许多与框架类似的性质,但是并不是所有的性质都是相似的.例如,无冗框架等价于Riesz基,但是无冗g-框架不等价于g-Riesz基.一些作者将Hilbert空间中的框架和对偶框架的等式和不等式推广到g-框架和对偶g-框架.本文建立Hilbert空间中的g-Bessel序列或g-框架的一些新的等式和不等式.本文还给出这些不等式的等号成立的充要条件.这些结果推广和改进了由Balan,Casazza和Gavruta等得到的著名结果.更多还原

【Abstract】 G-frames, which include many generalizations of frames such as frames of subspaces or fusion frames, oblique frames, and pseudo-frames, are natural generalizations of frames in Hilbert spaces. They have some properties similar to those of frames in Hilbert spaces, but not all of their properties are similar. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases. Some authors have extended the equalities and inequalities for frames and dual frames to g-frames and dual g-frames in Hilbert spaces. In this paper, we establish some new equalities and inequalities for g-Bessel sequences or g-frames in Hilbert spaces. We also give a necessary and sufficient condition that the equality occurs in one of these inequalities. Our results generalize and improve the remarkable results which had been obtained by Balan, Casazza and Gavruta.更多还原