Complex powers of operators. (English) Zbl 1261.47024

Summary: We define the complex powers of a densely defined operator \(A\) whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists \(\alpha\in[0,\infty)\) such that the resolvent of \(A\) is bounded by \(O((1+| \lambda|)^\alpha)\) there. We prove that, for some particular choices of a fractional number \(b\), the negative of the fractional power \((-A)^b\) is the c.i.g. of an analytic semigroup of growth order \(r>0\).


47A60 Functional calculus for linear operators
47D06 One-parameter semigroups and linear evolution equations
47D09 Operator sine and cosine functions and higher-order Cauchy problems
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