Maximum of the resolvent over matrices with given spectrum

详细信息    查看全文

• 作者：Oleg Szehr^a ; ^{oleg.szehr@posteo.de" class="auth_mail" title="E-mail the corresponding author} ; Rachid Zarouf^b ; ^c ; ^{rachid.zarouf@univ-amu.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 15A60 ; secondary ; 30D55
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：819-847
• 全文大小：487 K

文摘

In numerical analysis it is often necessary to estimate the condition number 8ab8b6edade2eb7ca078488e0dbd6267" title="Click to view the MathML source">CN(T)=‖T‖⋅‖T⁻¹‖$CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert$ and the norm of the resolvent e897e140cd1a4f40764b7a74bbf7a59d" title="Click to view the MathML source">‖(ζ−T)⁻¹‖$\Vert {(\zeta -T)}^{-1}\Vert$ of a given n×n$n\times n$ matrix T  . We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the fact that the supremum of 8a54aa25b29ff658610c746b9" title="Click to view the MathML source">CN(T)$CN(T)$ over all matrices with afa2eaf81181dca3df389da" title="Click to view the MathML source">‖T‖≤1$\Vert T\Vert \le 1$ and minimal absolute eigenvalue r=min_λ∈σ(T)⁡|λ|>0$r={\min }_{\lambda \in \sigma (T)}\left|\lambda \right|>0$ is the Kronecker bound e66b2f922ae84f5e">$\frac{1}{r^n}$. This result is subsequently generalized by computing for given ζ   in the closed unit disc the supremum of e897e140cd1a4f40764b7a74bbf7a59d" title="Click to view the MathML source">‖(ζ−T)⁻¹‖$\Vert {(\zeta -T)}^{-1}\Vert$, where afa2eaf81181dca3df389da" title="Click to view the MathML source">‖T‖≤1$\Vert T\Vert \le 1$ and the spectrum e6d1a842727a620e4816cfc0b00aaeb4" title="Click to view the MathML source">σ(T)$\sigma (T)$ of T   is constrained to remain at a pseudo-hyperbolic distance of at least ada40f8fd6b9e144f" title="Click to view the MathML source">r∈(0,1]$r\in (0,1]$ around ζ  . We find that the supremum is attained by a triangular Toeplitz matrix. This provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically. The occurring Toeplitz matrices are so-called model matrices, i.e. matrix representations of the compressed backward shift operator on the Hardy space e6463eb2fdf459e787fb344e26467" title="Click to view the MathML source">H₂$H_2$ to a finite-dimensional invariant subspace.