Isolated spectral points

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 0 is isolated in the spectrum of a bounded linear operator T T if the (not necessarily closed) space M = { x : lim n ‖ T n x ‖ 1 / n = 0 } M=\{x: \lim _{n}\|T^nx\|^{1/n}=0\} is nonzero and complemented by a closed subspace N N satisfying T N ⊂ N ⊂ T X TN\subset N\subset TX .