论文标题
可分解的弱操作员可值帧
Factorable Weak Operator-Valued Frames
论文作者
论文摘要
令$ \ MATHCAL {H} $和$ \ MATHCAL {H} _0 $为HILBERT SPACES，$ \ {a_n \} _ N $是$ \ Mathcal {h} $ to $ \ nathcal {h nathcal {h} _0 $ _0 $的序列。希尔伯特空间的研究框架启动了$ \ sum_ {n = 1}^{\ infty} a_n^*a_n $的运算符的研究，其中收敛是在强大操作器拓扑中，作者：Kaftal，larson and larson and larson and Zhang in Paper中：操作员的框架。 \ textit {trans。阿米尔。数学。 Soc。}，361（12）：6349-6385，2009。在本文中，我们将其概括并研究$ \ sum_ {n = 1}^{\ in = 1}^{\ infty}ψ_n^*a_n $的系列，其中$ \ \ \ \ \ \ {ψ_n\} _ n $是$ a $ a $的$ \ seque from $ { $ \ MATHCAL {H} _0 $。 $ \ sum_ {n = 1}^{\ infty} a_n^*a_n $的研究中使用的主要工具是本系列的分解。由于$ \ sum_ {n = 1}^{\ infty}ψ_n^*a_n $可能不会被考虑，因此需要更大的护理。因此，我们将$ \ sum_ {n = 1}^{\ infty}ψ_n^*a_n $的分解化。我们表征它们并得出扩张结果。我们通过将索引集作为组以及类似组的单一系统来进一步研究该系列。我们还得出稳定性结果。
Let $\mathcal{H}$ and $\mathcal{H}_0$ be Hilbert spaces and $\{A_n\}_n$ be a sequence of bounded linear operators from $\mathcal{H}$ to $\mathcal{H}_0$. The study frames for Hilbert spaces initiated the study of operators of the form $\sum_{n=1}^{\infty}A_n^*A_n$, where the convergence is in the strong-operator topology, by Kaftal, Larson and Zhang in the paper: Operator-valued frames. \textit{Trans. Amer. Math. Soc.}, 361(12):6349-6385, 2009. In this paper, we generalize this and study the series of the form $\sum_{n=1}^{\infty}Ψ_n^*A_n$, where $\{Ψ_n\}_n$ is a sequence of operators from $\mathcal{H}$ to $\mathcal{H}_0$. Main tool used in the study of $\sum_{n=1}^{\infty}A_n^*A_n$ is the factorization of this series. Since the series $\sum_{n=1}^{\infty}Ψ_n^*A_n$ may not be factored, it demands greater care. Therefore we impose a factorization of $\sum_{n=1}^{\infty}Ψ_n^*A_n$ and derive various results. We characterize them and derive dilation results. We further study the series by taking the indexed set as group as well as group-like unitary system. We also derive stability results.
