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Journal of the London Mathematical Society (1)

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WEYL'S THEOREM, $a$-WEYL'S THEOREM, AND LOCAL SPECTRAL THEORY
RAÚL E. CURTO, YOUNG MIN HAN
Journal:

Journal of the London Mathematical Society / Volume 67 / Issue 2 / April 2003

Published online by Cambridge University Press:

25 March 2003, pp. 499-509

Print publication:

April 2003
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Necessary and sufficient conditions are given for a Banach space operator with the single-valued extension property to satisfy Weyl's theorem and $a$-Weyl's theorem. It is shown that if $T$ or $T^{\ast}$ has the single-valued extension property and $T$ is transaloid, then Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma (T))$. When $T^{\ast}$ has the single-valued extension property, $T$ is transaloid and $T$ is $a$-isoloid, then <formula form="inline" disc="math" id="ffm012"><formtex notation="AMSTeX">$a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$. It is also proved that if $T$ or $T^{\ast}$ has the single-valued extension property, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.