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How can the following be defined? i) the spectrum of two linear operators in B(X) where B(X) is an algebra of Bounded of linear operators?

ii) essential numerical range of the above mentioned linear operators?

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Why do you expect these should be reasonable notions? –  Rasmus Aug 21 '12 at 11:29
In fact this is a difficult question. A sensible definition was given by J.L. Taylor in 1970 in his paper "A joint spectrum for several commuting operators". See this article for details –  Norbert Aug 21 '12 at 14:22

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