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Spectral Theory and Differential Operators$
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David Edmunds and Des Evans

Print publication date: 2018

Print ISBN-13: 9780198812050

Published to Oxford Scholarship Online: September 2018

DOI: 10.1093/oso/9780198812050.001.0001

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Entropy Numbers, s-Numbers, and Eigenvalues

Entropy Numbers, s-Numbers, and Eigenvalues

Chapter:
(p.43) 2 Entropy Numbers, s-Numbers, and Eigenvalues
Source:
Spectral Theory and Differential Operators
Author(s):

D. E. Edmunds

W. D. Evans

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198812050.003.0002

The geometric quantities entropy numbers, approximation numbers and n-widths are defined for compact linear maps, and connections with the analytic entities eigenvalues and essential spectra discussed. The celebrated inequality of Weyl between the approximation numbers and eigenvalues is established in the general context of Lorentz sequence spaces. Also included are an axiomatic approach to s-numbers, a discussion of non-compact maps, and the Schmidt decomposition theory for compact linear operators in Hilbert spaces.

Keywords:   approximation number, eigenvalue, entropy number, Weyl inequality, Lorentz space, s-number

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