Local quadratic estimates and holomorphic functional calculi. (English) Zbl 1226.47016

Hassell, Andrew (ed.) et al., The AMSI-ANU workshop on spectral theory and harmonic analysis. Proceedings of the workshop, Canberra, Australia, July 13–17, 2009. Canberra: Australian National University, Centre for Mathematics and its Applications (ISBN 978-0-7315-5208-5). Proceedings of the Centre for Mathematics and its Applications, Australian National University 44, 211-231 (2010).
Summary: We construct holomorphic functional calculi and introduce local quadratic estimates for operators in a reflexive Banach space that are bisectorial except possibly in a neighbourhood of the origin. The main result is an equivalence of local quadratic estimates with bounded holomorphic functional calculi. For operators with spectrum in a neighbourhood of the origin, the results are weaker than those for bisectorial operators. For operators with a spectral gap in a neighbourhood of the origin, the results are stronger. In each case, however, local quadratic estimates are a more appropriate tool than standard quadratic estimates for establishing that our functional calculi are bounded. This shows that in certain applications it suffices to establish local quadratic estimates.
For the entire collection see [Zbl 1218.47003].


47A60 Functional calculus for linear operators
47B44 Linear accretive operators, dissipative operators, etc.