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Joint spectra of commuting normal operators on Banach spaces. (English) Zbl 0689.47001
Let $$T=(T_ 1,...,T_ n)$$ be an n-tuple of operators on a complex Banach space X. T is called strongly commuting if there are operators $$U_ j$$, $$V_ j$$ with real spectra such that $$T_ j=U_ j+iV_ j$$ and $$(U_ 1,...,U_ n,V_ 1,...,V_ n)$$ is a commuting 2n-tuple. The author proves that the Taylor joint spectrum $$\sigma$$ (T) of a strong commuting n-tuple coincides with the approximate point spectrum $$\sigma_{\pi}(T)$$. For a commuting n-tuple T of normal operators, which is strongly commuting by the Fuglede theorem, the author proves moreover that co($$\sigma$$ (T)) is equal to the closure of the spatial joint numerical range $$\overline{V(T)}$$. Another result is that an extreme point $$z\in \overline{V(T)}$$ with $$z\in V(T)$$ is a joint eigenvalue of T.
Reviewer: M.Lesch

MSC:
 47A10 Spectrum, resolvent 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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