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Isolated spectral points. (English) Zbl 0864.46028
Summary: The paper studies isolated spectral points of elements of Banach algebras and of bounded linear operators in terms of the existence of idempotents, and gives an elementary characterization of spectral idempotents. It is shown that $$0$$ is isolated in the spectrum of a bounded linear operator $$T$$ if the (not necessarily closed) space $$M=\{x: \lim_{n}\| T^nx\|^{1/n}=0\}$$ is nonzero and complemented by a closed subspace $$N$$ satisfying $$TN\subset N\subset TX$$.

##### MSC:
 46H30 Functional calculus in topological algebras 47A10 Spectrum, resolvent 47A60 Functional calculus for linear operators
##### Keywords:
isolated spectral points; spectral idempotents
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##### References:
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