Semigroups and resolvents of bounded variation imaginary powers and \(H^ \infty\) functional calculus. (English) Zbl 0779.47036
Summary: Let \(-A\) be a linear, injective operator, on a Banach space \(X\). We show that an \(H^ \infty\) functional calculus for \(A\) exists if and only if \(-A\) generates a bounded strongly continuous holomorphic semigroup of uniform weak bounded variation, if and only if \(A(z+A)^{-1}\) is of uniform weak bounded variation. This provides a sufficient condition for the imaginary powers of \(A\), \(\{A^{-is}\}_{s\in\mathbb{R}}\), to extend to a strongly continuous group of bounded operators; we also give similar necessary conditions.
MSC:
| 47D06 | One-parameter semigroups and linear evolution equations |
| 46H30 | Functional calculus in topological algebras |
| 47A60 | Functional calculus for linear operators |
Keywords:
functional calculus; bounded strongly continuous holomorphic semigroup of uniform weak bounded variationReferences:
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