A simultaneous lifting theorem for block diagonal operators

Allen, G. D.; Ward, J. D.

doi:10.1090/S0002-9947-1979-0530063-8

A simultaneous lifting theorem for block diagonal operators
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by G. D. Allen and J. D. Ward PDF

Trans. Amer. Math. Soc. 250 (1979), 385-397 Request permission

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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