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 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 e8517b6df45973531d386d71e3c7f04" 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 e8517b6df45973531d386d71e3c7f04" 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 83b9a04f6fc58f6d1f4b1fa3652a0" title="Click to view the MathML source">{T_n}$\{T_n\}$ of finite matrices such that 9c1e4eae7" title="Click to view the MathML source">Ψ_Tn(z)${\mathrm{\Psi }}_{T_n}(z)$ tends to e71527ff2f56373a" title="Click to view the MathML source">Ψ_T(z)${\mathrm{\Psi }}_T(z)$ uniformly on 83356cc41b4a314c6dbc" title="Click to view the MathML source">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.

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 9c4953a0afda6483b1e09bb437d3a65a" title="Click to view the MathML source">Ψ_S${\mathrm{\Psi }}_S$, the function 9c00210a41bb474fe248be6b1e">$\Vert \sqrt{S-z}\Vert$ for certain finite matrices S may oscillate arbitrarily fast even far away from the spectrum.