算子Riccati方程解的存在唯一性。几何方法
Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach
摘要：
本文对Riccati算子方程$A_1 X - X A_0 - X V X + V^\ast = 0$引入了无界解的新概念，并给出了与分块算子矩阵$\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1 \end{pmatrix}$的谱图子空间相关的解的完整描述.我们还提供了一个新的特征的集合的所有压缩的解决方案的假设下，Riccati方程有一个压缩的解决方案与一个谱子空间的操作$\mathbf{B}$。在这种情况下，我们建立了压缩解的唯一性准则。
英文摘要：
We introduce a new concept of unbounded solutions to the operator Riccati equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix $\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1 \end{pmatrix}$. We also provide a new characterization of the set of all contractive solutions under the assumption that the Riccati equation has a contractive solution associated with a spectral subspace of the operator $\mathbf{B}$. In this case we establish a criterion for the uniqueness of contractive solutions.
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。
