Title:

Operator logarithms and exponentials

Since Mclntosh's introduction of the H^{∞}calculus for sectorial operators, the topic has been studied by many authors. Haase has constructed a similar functional calculus for striptype operators, and has also developed an abstract framework which unifies both of these examples and more. In this thesis we use this abstract functional calculus setting to study two particular problems in operator theory. The first of these is concerned with operator sums. We ask the question of when the sum log A+log B is closed, where A and B are a pair of injective sectorial operators whose resolvents commute. We show that the sum is always closable and, when A and B are invertible, we determine sufficient conditions for the sum to be closed. These conditions are of KaltonWeis type, and in fact ensure that AB is sectorial and that the identity log A + log B = log(AB) holds. We then identify an interpolation space on which these conditions are automatically satisfied. Our second problem is connected to the exponential of a striptype operator B, specifically the question of whether e^{B} is sectorial. When 1 ∈ p(e^{B}), the spectrum of e^{B} lies in a sector, and we obtain an estimate on the resolvent outside this sector. This estimate becomes closer to sectoriality as more restrictions are placed on the resolvents of B itself. This leads us to introduce the ideas of Fsectorial and Fstrong striptype operators, whose spectra are contained in a sector or strip, but which satisfy a different resolvent estimate from that of a sectorial or strong striptype operator. In some cases it is possible to define the logarithm of an Fsectorial operator or the exponential of an Fstrong striptype operator. We prove resolvent estimates for the resulting logarithms and exponentials, and explore the relationships between the various classes of operators considered.
