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Streszczenie

Let $\phi (z)$ be an analytic function in a disk $|z| < \rho $ (in particular, a polynomial) such that $\phi (0)=1$, $\phi (z)\not \equiv 1$. Let $V$ be the operator of integration in $L_p(0,1)$, $1\leq p\leq \infty $. Then $\phi (V)$ is power bounded if and only if $\phi '(0)<0$ and $p=2$. In this case some explicit upper bounds are given for the norms of $\phi (V)^n$ and subsequent differences between the powers. It is shown that $\phi (V)$ never satisfies the Ritt condition but the Kreiss condition is satisfied if and only if $\phi '(0)<0$, at least in the polynomial case.