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26 April 2024 |
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Sharpening a result by E.B. Davies and B. Simon | | Rachid Zarouf
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16 Mar 2009 | | Abstract: | E. B. Davies et B. Simon have shown (among other things) the following
result: if T is an n imes n matrix such that its spectrum sigma(T) is
included in the open unit disc mathbb{D}={zinmathbb{C}: | z|<1} and if
C=sup_{kgeq0}Vert T^{k}Vert_{E o E}, where E stands for mathbb{C}^{n}
endowed with a certain norm |.|, then Vert R(1, T)Vert_{E o E}leq
C(3n/dist(1, sigma(T)))^{3/2} where R(lambda, T) stands for the resolvent of
T at point lambda. Here, we improve this inequality showing that under the
same hypotheses (on the matrix T), Vert R(lambda, T)Vert leq
C(5pi/3+2sqrt{2})n^{3/2}/dist(lambda, sigma), for all
lambda
otinsigma(T) such that |lambda|geq1. | | Source: | arXiv, 0903.2743 | | Services: | Forum | Review | PDF | Favorites | |
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