弱形式Sylvester方程及其应用
On weakly formulated Sylvester equations and applications
AI总结：
利用弱形式Sylvester方程得到Hilbert空间中非负自伴算子谱子空间旋转的新界，推广了Davis-Kahan结果，并给出平方根算子的误差界。
摘要：
我们使用一个“弱形式”的Sylvester方程 $$A^{1/2}TM^{-1/2}-A^{-1/2}TM^{1/2}=F$$ 来获得Hilbert空间中非负自伴算子的谱子空间旋转的新界。我们的界推广了Davis和Kahan的已知结果。另一个应用是正自伴算子平方根的界，它推广了已知规则：“平方根中的相对误差被根号中相对误差的一半所界”。这两个界都在通过二次型定义的微分算子上进行了说明。
英文摘要：
We use a ``weakly formulated'' Sylvester equation $$A^{1/2}TM^{-1/2}-A^{-1/2}TM^{1/2}=F$$ to obtain new bounds for the rotation of spectral subspaces of a nonnegative selfadjoint operator in a Hilbert space. Our bound extends the known results of Davis and Kahan. Another application is a bound for the square root of a positive selfadjoint operator which extends the known rule: ``The relative error in the square root is bounded by the one half of the relative error in the radicand''. Both bounds are illustrated on differential operators which are defined via quadratic forms.
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