论文标题
Schur补充主导的操作员矩阵
Schur complement dominant operator matrices
论文作者
论文摘要
我们提出了一种在一般环境中无界运算符矩阵的光谱分析的方法，该矩阵完全戒除了标准的扰动参数。我们不需要矩阵在希尔伯特空间$ \ Mathcal {h} $中起作用，而是将其操作扩展到合适的分布三重$ \ Mathcal {d} \ subset \ subset \ Mathcal \ Mathcal {h} \ subset \ subset \ subset \ Mathcal \ Mathcal {d} _- $，并将其限制到其Maximal domain in $ \ Mathain in $ \ \ \ = $ {我们方法中的关键点是选择空间$ \ MATHCAL {D} $和$ \ MATHCAL {D} _- $的选择，这实际上是由矩阵的Schur补充确定的。我们在$ \ Mathcal {H} $中显示了所得的运算符矩阵与其Schur补体之间的光谱等效性，该矩阵可以从Schur补充的合适表示（例如通过广义形式方法）传递到操作员矩阵的表示。因此，我们概括了经典光谱等效性结果，构成了标准优势模式。 抽象结果应用于具有无界和/或奇异阻尼的阻尼波方程式，具有具有库仑型电势的狄拉克运算符，以及通用的二阶矩阵差异差异操作员。通过我们的方法，可以实质上削弱以前的规律性假设。
We propose a method for the spectral analysis of unbounded operator matrices in a general setting which fully abstains from standard perturbative arguments. Rather than requiring the matrix to act in a Hilbert space $\mathcal{H}$, we extend its action to a suitable distributional triple $\mathcal{D} \subset \mathcal{H} \subset \mathcal{D}_-$ and restrict it to its maximal domain in $\mathcal{H}$. The crucial point in our approach is the choice of the spaces $\mathcal{D}$ and $\mathcal{D}_-$ which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in $\mathcal{H}$ and its Schur complement, which allows to pass from a suitable representation of the Schur complement (e.g. by generalised form methods) to a representation of the operator matrix. We thereby generalise classical spectral equivalence results imposing standard dominance patterns. The abstract results are applied to damped wave equations with possibly unbounded and/or singular damping, to Dirac operators with Coulomb-type potentials, as well as to generic second order matrix differential operators. By means of our methods, previous regularity assumptions can be weakened substantially.
