Operator logarithms and exponentials - Clark, S; Clark, Stephen (2007)
﻿ Since Mclntosh's introduction of the 𝛨∞-calculus for sectorial operators, the topic has been studied by many authors. Haase has constructed a similar functional calculus for strip-type 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 Kalton-Weis 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 strip-type operator B , specifically the question of whether eB is sectorial. When -1 ∈ p(eB), the spectrum of eB 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 F-sectorial and F-strong strip-type 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 F-sectorial operator or the exponential of an F-strong strip-type operator. We prove resolvent estimates for the resulting logarithms and exponentials, and explore the relationships between the various classes of operators considered.