算子Riccati方程解的存在性及tanΘ定理
On the existence of solutions to the operator Riccati equation and the tanΘtheorem
摘要：
设A和C是自伴算子，使得A的谱位于C的谱的间隙中，d>0是A和C的谱之间的距离.我们证明，在这些假设下，常数c的尖锐值在条件||B||保证算子Riccati方程XA-CX+XBX=B^* 的（有界）解的存在性的<cd等于\sqrt{2}。我们还证明了Davis-Kahan \tan Θ定理的一个推广，并给出了Riccati方程解的模的精确估计.如果C是有界的，我们证明，此外，解X是严格压缩，如果B满足条件||B|| <d，这个条件是尖锐的。
英文摘要：
Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d>0 be the distance between the spectra of A and C. We prove that under these assumptions the sharp value of the constant c in the condition ||B||<cd guaranteeing the existence of a (bounded) solution to the operator Riccati equation XA-CX+XBX=B^* is equal to \sqrt{2}. We also prove an extension of the Davis-Kahan \tanΘtheorem and provide a sharp estimate for the norm of the solution to the Riccati equation. If C is bounded, we prove, in addition, that the solution X is a strict contraction if B satisfies the condition ||B||<d, and that this condition is sharp.
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