# 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
$\left \| \left ( B-\lambda I \right )^{-1} \right \|\leq \varphi _{B}(\frac{1}{\left | \lambda \right |})$ $(\left | \lambda \right |> 1)$,
where $\varphi _{B}\left ( z \right )=\sum_{1}^{\infty }\left \| B^{n-1} \right \|z^{n}$ is an analytical function on the unit disc $\mathbb{D}=\left \{ \lambda :\left | \lambda \right | < 1\right \}$.
Describing the growth of the resolvent and the behavior of powers of operators and also the relation between them are classical problems in spectral theory of linear operators.
One of the most known results discussed the behavior of function $\varphi _{B}\left ( z \right )$ was obtained by H.-O. Kreiss in [2]. There were shown at the finite-dimentional space that if the spectrum is belongs to the unit disc and the resolvent for $\left | \lambda \right |> 1$ holds
$\left \| \left ( B-\lambda I \right )^{-1} \right \|\leq \frac{C}{\left | \lambda \right |-1}$,
then the operator is power-bounded i.e.
$\sup_{n\geq 0}\left \| B^{n} \right \|=C< \infty$
and this proposition is not true at the infinite-dimentional space.
In this paper we considered more rapidly growth for the resolvent which we called generalized Kriess $\left ( \gamma ,\rho \right )$ -condition such that
$\left \| \left ( B-\lambda I \right )^{-1} \right \|\leq C\exp\frac{\rho }{\left ( \left | \lambda \right | -1\right )\gamma }$,
where $C$ is constant and $\gamma ,\rho$ are the order and type of the growth of the resolvent, respectively .
For an arbitrary operator $B$ the relation between the mentioned above condition and the behavior of power of operator is obtained. As an example, for the discrete weighted shift operators the relation between the behavior of coefficient and the fulfilment of generalized Kriess $\left ( \gamma ,\rho \right )$ -condition is given.

Keywords: resolvent, Kriess resolvent condition, power-bounded operator, analytical function on the disc, discrete weighed shift operators.

UDC:
517.984