论文标题
关于希尔伯特空间紧凑型正常操作员的Krylov溶解度的注释
A note on the Krylov solvability of compact normal operators on Hilbert space
论文作者
论文摘要
我们分析了希尔伯特空间上的逆线性问题的Krylov溶解度$ \ Mathcal {H} $，其中底层操作员紧凑且正常。 Krylov的溶解度是逆线性问题的重要特征，它在理论和应用数值分析中具有深远的影响，因为了解基于Krylov的实用性以解决反向问题的实用性至关重要。我们的结果首次明确描述了该操作员的Krylov子空间，因为\ Mathcal {H} $中的任何基准矢量$ g \ g \ in \ Mathcal {h} $，并证明所有倒线性问题都是Krylov溶液的，只要$ G $在此类操作员的范围内。因此，我们将对Krylov可解决的操作员类别的知识扩展到包括正常紧凑型操作员。我们通过证明通用正常运算符的封闭的Krylov子空间与基于标量光谱度量的$ l^2 $量化空间之间的同构合法性。
We analyse the Krylov solvability of inverse linear problems on Hilbert space $\mathcal{H}$ where the underlying operator is compact and normal. Krylov solvability is an important feature of inverse linear problems that has profound implications in theoretical and applied numerical analysis as it is critical to understand the utility of Krylov based methods for solving inverse problems. Our results explicitly describe for the first time the Krylov subspace for such operators given any datum vector $g\in\mathcal{H}$, as well as prove that all inverse linear problems are Krylov solvable provided that $g$ is in the range of such an operator. We therefore expand our knowledge of the class of Krylov solvable operators to include the normal compact operators. We close the study by proving an isomorphism between the closed Krylov subspace for a general bounded normal operator and an $L^2$-measure space based on the scalar spectral measure.
