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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An inequality for some nonnormal operators


Author: Takayuki Furuta
Journal: Proc. Amer. Math. Soc. 104 (1988), 1216-1217
MSC: Primary 47A30; Secondary 47B20, 65J10
DOI: https://doi.org/10.1090/S0002-9939-1988-0969053-0
MathSciNet review: 969053
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Abstract: An inequality of use in testing convergence of eigenvector calculations is improved. If $ {e_\lambda }$ is a unit eigenvector corresponding to an eigenvalue $ \lambda $ of a dominant operator $ A$ on a Hilbert space $ H$, then

$\displaystyle \vert(g,{e_\lambda }){\vert^2} \leq \frac{{\vert\vert g\vert{\ver... ...t{\vert^2} - \vert(g,Ag){\vert^2}}}{{\vert\vert(A - \lambda I)g\vert{\vert^2}}}$

for all $ g$ in $ H$ for which $ Ag \ne \lambda g$. The equality holds if and only if the component of $ g$ orthogonal to $ {e_\lambda }$ is also an eigenvector of $ A$. This result is an improvement of Bernstein's result for selfadjoint operators.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0969053-0
Keywords: Eigenvector, dominant operator
Article copyright: © Copyright 1988 American Mathematical Society