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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si1.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=8ab8b6edade2eb7ca078488e0dbd6267" title="Click to view the MathML source">CN(T)=‖T‖⋅‖T⁻¹‖class="mathContainer hidden">class="mathCode">$CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert$ and the norm of the resolvent class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si2.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=e897e140cd1a4f40764b7a74bbf7a59d" title="Click to view the MathML source">‖(ζ−T)⁻¹‖class="mathContainer hidden">class="mathCode">$\Vert {(\zeta -T)}^{-1}\Vert$ of a given class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si3.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=6d7946b5ef60609520823ee254fb547e" title="Click to view the MathML source">n×nclass="mathContainer hidden">class="mathCode">$n\times n$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si4.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=4bee9478a54aa25b29ff658610c746b9" title="Click to view the MathML source">CN(T)class="mathContainer hidden">class="mathCode">$CN(T)$ over all matrices with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si5.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=0453cc90dafa2eaf81181dca3df389da" title="Click to view the MathML source">‖T‖≤1class="mathContainer hidden">class="mathCode">$\Vert T\Vert \le 1$ and minimal absolute eigenvalue class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si6.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=42f6c4e7c5d76e2729a11ae95cc5b7cf" title="Click to view the MathML source">r=min_λ∈σ(T)⁡|λ|>0class="mathContainer hidden">class="mathCode">$r={\min }_{\lambda \in \sigma (T)}\left|\lambda \right|>0$ is the Kronecker bound class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si7.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=11e178d62c775432e66b2f922ae84f5e">class="imgLazyJSB inlineImage" height="20" width="18" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616301938-si7.gif">class="mathContainer hidden">class="mathCode">. This result is subsequently generalized by computing for given ζ   in the closed unit disc the supremum of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si2.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=e897e140cd1a4f40764b7a74bbf7a59d" title="Click to view the MathML source">‖(ζ−T)⁻¹‖class="mathContainer hidden">class="mathCode">$\Vert {(\zeta -T)}^{-1}\Vert$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si5.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=0453cc90dafa2eaf81181dca3df389da" title="Click to view the MathML source">‖T‖≤1class="mathContainer hidden">class="mathCode">$\Vert T\Vert \le 1$ and the spectrum class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si8.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=e6d1a842727a620e4816cfc0b00aaeb4" title="Click to view the MathML source">σ(T)class="mathContainer hidden">class="mathCode">$\sigma (T)$ of T   is constrained to remain at a pseudo-hyperbolic distance of at least class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si148.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=3bce0389eaa5e7eada40f8fd6b9e144f" title="Click to view the MathML source">r∈(0,1]class="mathContainer hidden">class="mathCode">$r\in (0,1]$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301938&_mathId=si10.gif&_user=111111111&_pii=S0022123616301938&_rdoc=1&_issn=00221236&md5=28fe6463eb2fdf459e787fb344e26467" title="Click to view the MathML source">H₂class="mathContainer hidden">class="mathCode">$H_2$ to a finite-dimensional invariant subspace.