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Abstract

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.