On the Gelfand-Hille theorems

@inproceedings{Zemnek1994OnTG,
  title={On the Gelfand-Hille theorems},
  author={Jaroslav Zem{\'a}nek},
  year={1994}
}
Let T be a bounded linear operator on a complex Banach space X, with smallest possible spectrum, say, σ(T ) = {1}. Thus, the resolvent (T −λI)−1 is an analytic function of λ on C \ {1}, vanishing at infinity, and the point 1 is either a pole or an essential singularity. More precisely, it is a pole of order r if and only if r is the least exponent such that (T − I) = 0, because (T − λI)−1 = −I(λ− 1)−1− (T − I)(λ− 1)−2− . . .− (T − I)(λ− 1)−(n+1)− . . . 

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