Title:

The bounded H∞ calculus for sectorial, striptype and halfplane operators

The main study of this thesis is the holomorphic functional calculi for three classes of unbounded operators: sectorial, striptype and halfplane. The functional calculus for sectorial operators was introduced by McIntosh as an extension of the RieszDunford model for bounded operators. More recently Haase has developed an abstract framework which incorporates analogous constructions for striptype and halfplane operators. These operators are of interest since they arise naturally as generators of C_{0}(semi)groups. The theory of bounded H^{∞}calculus for sectorial operators is well established and it has been found to have many applications in operator theory and parabolic evolution equations. We survey these known results, first on Hilbert space and then on general Banach space. Our main goal is to fill the gaps in the parallel theory for striptype operators. Whilst some of this can be deduced by taking exponentials and applying known results for sectorial operators, in general this is insu_cient to obtain our desired results and so we pursue an independent approach. Starting on Hilbert space, we broaden known characterisations of the bounded H^{∞}calculus for striptype operators by introducing a notion of absolute calculus which is an analogue to the established notion for the sectorial case. Moving to general Banach space, we build on the work of Vörös, broadening his characterisation for striptype operators in terms of weak integral estimates by introducing a new, but equivalent, notion of the bounded H^{∞}calculus, which we call the mbounded calculus. We also demonstrate that these characterisations fail for halfplane operators and instead present a weaker form of the bounded Hcalculus which is more natural for these operators. This allows us to obtain new and simple proofs of well known generation theorems due to Gomilko and ShiFeng, with extensions to polynomially bounded semigroups. The connection between the bounded Hcalculus of semigroup generators and polynomial boundedness of their associated Cayley Transforms is also explored. Finally we present a series of results on sums of operators, in connection with maximal regularity. We also establish stability results for the bounded H^{∞}calculus for striptype operators by showing it is preserved under suitable bounded perturbations, which at time requires further assumptions on the underlying Banach space. This relies heavily on intermediate characterisations of the bounded H^{∞}calculus due to Kalton and Weis.
