框架序列算子及其伪逆的系统研究
A Systematic Study of Frame Sequence Operators and their Pseudoinverses
摘要：
本文研究Hilbert空间$H$中与框架序列相关的算子，即，合成算子$T:\ell ^{2}(\mathbb{N}) \to H$、分析算子$T^{\ast}:H\to $ $% \ell ^{2}(\mathbb{N}) $和相关的框架算子$S=TT^{\ast}$作为定义在整个空间而不是子空间上的算子。进一步研究了$P$在$T$值域上的投影、$Q$在$T^{\ast}$值域上的投影以及Gram矩阵$G=T^{\ast}T$.对于所有这些运营商，我们调查他们的伪逆，他们如何相互作用，以及可能的分类框架序列与他们。对于一个紧框架序列，我们表明，这些运营商中的一些连接在一个简单的方式。
英文摘要：
In this note we investigate the operators associated with frame sequences in a Hilbert space $H$, i.e., the synthesis operator $T:\ell ^{2}(\mathbb{N}) \to H$, the analysis operator $T^{\ast}:H\to $ $% \ell ^{2}(\mathbb{N}) $ and the associated frame operator $S=TT^{\ast}$ as operators defined on (or to) the whole space rather than on subspaces. Furthermore, the projection $P$ onto the range of $T$, the projection $Q$ onto the range of $T^{\ast}$ and the Gram matrix $G=T^{\ast}T$ are investigated. For all these operators, we investigate their pseudoinverses, how they interact with each other, as well as possible classification of frame sequences with them. For a tight frame sequence, we show that some of these operators are connected in a simple way.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
