Asymptotic type for sectorial operators and an integral of fractional powers (Q1011430)

This paper introduces and studies the notion of asymptotic type of linear operators. A closed linear operator \(V\) on a Banach space is of asymptotic type \(\omega\) if the operator \(\lambda(\lambda I-V)^{-1}\) is bounded in certain neighourhoods of zero. If \(V\) is sectorial and of asymptotic type \(\omega\), then the fractional power \(V^\alpha\) is of asymptotic type \(\alpha\omega\) for a suitable range of positive \(\alpha\). The proof is a modification of Kato's proof of a similar statement for sectorial operators. Further, various properties of the operator \(\int_0^1 V^\alpha \,d\alpha\), where \(V\) is a sectorial operator, are studied. In particular, it is shown that \(\int_0^1 V^\alpha \,d\alpha\) is of asymptotic type \(0\). Using this result, an operator satisfying the Ritt resolvent condition is constructed.