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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Distance between normal operators


Author: V. S. Sunder
Journal: Proc. Amer. Math. Soc. 84 (1982), 483-484
MSC: Primary 47B15; Secondary 15A60, 47A55
MathSciNet review: 643734
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Abstract: Lidskii and Wielandt have proved independently that if $ A$ and $ B$ are selfadjoint operators on an $ n$-dimensional space $ H$, with eigenvalues $ \{ {\alpha _k}\} _{k = 1}^n$ and $ \{ {\beta _k}\} _{k = 1}^n$ respectively (counting multiplicity), then,

$\displaystyle \left\Vert {A - B} \right\Vert \geqslant \mathop {\min }\limits_{... ...{{\text{diag}}\left( {{\alpha _k} - {\beta _{\sigma (k)}}} \right)} \right\Vert$

for any unitarily invariant norm on $ L(H)$. In this note an example is given to show that this result is no longer true if $ A$ and $ B$ are only required to be normal (even unitary). It is also shown that the above inequality holds in the operator norm, if $ A$ is selfadjoint and $ B$ is skew-self-adjoint.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0643734-5
PII: S 0002-9939(1982)0643734-5
Article copyright: © Copyright 1982 American Mathematical Society