The power boundedness and resolvent conditions for functions of the classical Volterra operator

Yuri Lyubich ¹

¹ Institute of Mathematics Polish Academy of Sciences Warszawa, Poland and Technion, Haifa, Israel

Studia Mathematica, Tome 196 (2010) no. 1, pp. 41-63

Résumé

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.

Détail

DOI : 10.4064/sm196-1-4

Keywords: phi analytic function disk rho particular polynomial phi phi equiv operator integration leq leq infty phi power bounded only phi explicit upper bounds given norms phi subsequent differences between powers shown phi never satisfies ritt condition kreiss condition satisfied only phi least polynomial