On an integral of fractional power operators

Nick Dungey¹

¹ Department of Mathematics Macquarie University Sydney, NSW 2109, Australia

Colloquium Mathematicum, Tome 117 (2009) no. 2, pp. 157-164

Résumé

For a bounded and sectorial linear operator $V$ in a Banach space, with spectrum in the open unit disc, we study the operator $\widetilde{V} = \int_0^{\infty} d\alpha\, V^{\alpha}$. We show, for example, that $\widetilde{V}$ is sectorial, and asymptotically of type $0$. If $V$ has single-point spectrum $\{0\}$, then $\widetilde{V}$ is of type $0$ with a single-point spectrum, and the operator $I-\widetilde{V}$ satisfies the Ritt resolvent condition. These results generalize an example of Lyubich, who studied the case where $V$ is a classical Volterra operator.

Détail

DOI : 10.4064/cm117-2-1

Keywords: bounded sectorial linear operator banach space spectrum unit disc study operator widetilde int infty alpha alpha example widetilde sectorial asymptotically type has single point spectrum nbsp widetilde type single point spectrum operator i widetilde satisfies ritt resolvent condition these results generalize example lyubich who studied where classical volterra operator