Fractional powers of operators with polynomially bounded resolvent and the semigroups generated by them (Q1894981)

The paper defines fractional powers of type \((- A)^b\), where \(b\in \mathbb{C}\), the resolvent of \(A\) is bounded by a polynomial of order \(n\) and the resolvent set contains a closed sector in \(\mathbb{C}\) around the origin. The definition is an extension of an earlier one given by Fattorini an S. G. Krein in the case \(n= -1\). Motivated by applications to incomplete Cauchy problems for high order abstract differential equations, one shows that for \(0< b\leq {1\over 2}\) the operator \(- (- A)^b\) is the complete generator of an analytic semigroup of growth order \({n+ 1\over b}\).