Hilbert空间中的g-框架类展开式与g-Riesz-Fischer序列

g-frame-like expansions and g-Riesz-Fischer sequences in Hilbert spaces

本文考虑了Hilbert空间中一些g-框架类展开式以及g-Riesz-Fischer序列.首先讨论了Hilbert空间中一类算子序列,该类算子序列不必是g-Bessel序列.本文证明了在某些条件下g-框架类展开式存在并且这些展开式保持了通常意义下的能量最小性质.然后讨论了满足下g-框架条件的算子序列,这些算子序列不必满足上g-框架界,得到了Hilbert空间的子空间中的g-框架类展开式,并且给出了一个满足下g-框架条件的算子序列的一个刻画.最后讨论了Hilbert空间中的g-Riesz-Fischer序列,该类算子序列与满足下g-框架条件的算子序列紧密相关,得到了g-Riesz-Fischer序列的一些刻画.

In this paper, we consider some g-frame-like expansions in Hilbert spaces. Firstly, we consider the operator sequences which are not necessary g-Bessel sequences. We prove that under certain conditions a g-framelike expansion exists and this expansion maintains the minimal property as the usual case. Then we consider the operator sequences which satisfy the lower g-frame condition without assuming the upper g-frame bound. In this case, a g-frame-like expansion for a subspace is established. We also give a characterization for a sequence which satisfies the lower g-frame condition. Finally, we study the g-Riesz-Fischer sequences for Hilbert spaces which are closely related with the sequences which satisfy the lower g-frame condition. Some characterizations for g-Riesz-Fischer sequences are obtained.