Антоневич А. Б.

On operators with exponential growth of the resolvent.

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Let $B$ linear bounded operator and let spectral radius is $R\left ( B \right )=1$. Well-known that the resolvent operator can be represented by power series $\left ( B-\lambda I \right )^{-1}=-\sum_{n=0}^{\infty }\lambda ^{-n}B^{n-1}$ and the norm of the resolvent holds
$\left \| \left ( B-\lambda I \right )^{-1} \right \|\leq \varphi _{B}(\frac{1}{\left | \lambda \right |})$ $(\left | \lambda \right |> 1)$,