Infinite-dimensional features of matrices and pseudospectra

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• 作者：Avijit Pal^a ; ^{avijitmath@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Dmitry V. Yakubovich^b ; ^c ; ^{dmitry.yakubovich@uam.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Pseudospectra ; Quasitriangular ; Cowen&ndash ; Douglas class ; Nonnormal matrices ; Approximation of spectra ; Finite section method
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：109-127
• 全文大小：503 K

文摘

Given a Hilbert space operator T  , the level sets of function 0fde97b5d2" title="Click to view the MathML source">Ψ_T(z)=‖(T−z)⁻¹‖⁻¹${\mathrm{\Psi }}_T(z)={\Vert {(T-z)}^{-1}\Vert }^{-1}$ determine the so-called pseudospectra of T  . We set f45973531d386d71e3c7f04" title="Click to view the MathML source">Ψ_T${\mathrm{\Psi }}_T$ to be zero on the spectrum of T  . After giving some elementary properties of f45973531d386d71e3c7f04" title="Click to view the MathML source">Ψ_T${\mathrm{\Psi }}_T$ (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove that for any operator T  , there is a sequence {T_n}$\{T_n\}$ of finite matrices such that Ψ_Tn(z)${\mathrm{\Psi }}_{T_n}(z)$ tends to Ψ_T(z)${\mathrm{\Psi }}_T(z)$ uniformly on C$\mathbb{C}$. In this proof, quasitriangular operators play a special role. This is merely an existence result, we do not give a concrete construction of this sequence of matrices.

20">One of our main points is to show how to use infinite-dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size. In particular, we get a result, which means, in a sense, that the pseudospectrum of a nilpotent matrix can be anything one can imagine. We also study the norms of the multipliers in the context of Cowen–Douglas class operators. We use these results to show that, to the opposite to the function Ψ_S${\mathrm{\Psi }}_S$, the function $\Vert \sqrt{S-z}\Vert$ for certain finite matrices S may oscillate arbitrarily fast even far away from the spectrum.