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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A simultaneous lifting theorem for block diagonal operators


Authors: G. D. Allen and J. D. Ward
Journal: Trans. Amer. Math. Soc. 250 (1979), 385-397
MSC: Primary 47A10; Secondary 47A12, 47A55
MathSciNet review: 530063
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Abstract: Stampfli has shown that for a given $ T\, \in \,B\left( H \right)$ there exists a $ K\, \in \,C\left( H \right)$ so that $ \sigma \left( {T\, + \, K} \right)\,= \,{\sigma _w}\left( T \right)$. An analogous result holds for the essential numerical range $ {W_e}\left( T \right)$. A compact operator K is said to preserve the Weyl spectrum and essential numerical range of an operator $ T\, \in \,B\left( H \right)$ if $ \sigma \left( {T\, + \, K} \right)\, = \,{\sigma _w}\left( T \right)$ and $ \overline {W\left( {T \, + \, K} \right)} \, = \,{W_e}\left( T \right)$.

Theorem. For each block diagonal operator T, there exists a compact operator K which preserves the Weyl spectrum and essential numerical range of T.

The perturbed operator $ T \, + \, K$ is not, in general, block diagonal. An example is given of a block diagonal operator T for which there can be no block diagonal perturbation which preserves the Weyl spectrum and essential numerical range of T.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0530063-8
PII: S 0002-9947(1979)0530063-8
Keywords: Essential numerical range, Weyl spectrum, block diagonal operator
Article copyright: © Copyright 1979 American Mathematical Society