On Functional Calculus Estimates

This thesis presents various results within the field of operator theory that are formulated in estimates for functional calculi. Functional calculus is the general concept of defining operators of the form $f(A)$, where f is a function and $A$ is an operator, typically on a Banach space. Norm estimates for $f(A)$ emerge in applications naturally, e.g., when studying stability of numerical schemes. The first part deals with the $H^\infty$ functional calculus for generators $A$ of strongly continuous semigroups. Here, the functions $f$ are bounded and analytic on a half-plane in the complex plane. Within this part, we further distinguish the following topics. First, an alternative approach to define the $H^\infty$ calculus is presented. As a consequence, sufficient conditions for a bounded calculus are given. Second, we restrict to generators of analytic semigroups and allow for functions $f$ that are bounded and analytic on sectors. We show how the possible unboundedness of the calculus can be measured in terms of functional calculus estimates and describe how these estimates change through the occurrence of square function estimates. Third, we consider Tadmor--Ritt operators. The provided functional calculus estimates imply a new, simpler proof for the power-boundedness of Tadmor--Ritt operators and yield the best-known bound. Furthermore, the influence of discrete square function estimates is described. Fourth, the question whether the Cayley transform of the generator is power-bounded if the corresponding semigroup is bounded, is studied. Finding the answer in the case of semigroups on Hilbert spaces has become an enigmatic open problem in the last decades. Using a well-known link with the Inverse Generator Problem, we prove that -- to find the answer -- it suffices to consider exponentially stable semigroups. Moreover, it is shown that the Cayley Transform Problem and the Inverse Generator Problem are equivalent in a general sense. The second part of the thesis is on cosine families, which can be seen as the analog of semigroups for second-order Cauchy problems. We prove so-called zero-two laws for cosine families on Banach spaces and normed algebras, which have been open so far