Abstract
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 halfplane 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 TadmorRitt operators. The provided functional calculus estimates imply a new, simpler proof for the powerboundedness of TadmorRitt operators and yield the bestknown bound. Furthermore, the influence of discrete square function estimates is described.
Fourth, the question whether the Cayley transform of the generator is powerbounded 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 wellknown 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 secondorder Cauchy problems. We prove socalled zerotwo laws for cosine families on Banach spaces and normed algebras, which have been open so far.
Original language  Undefined 

Awarding Institution 

Supervisors/Advisors 

Thesis sponsors  
Award date  25 Sep 2015 
Place of Publication  Enschede 
Publisher  
Print ISBNs  9789036539623 
DOIs  
Publication status  Published  25 Sep 2015 
Keywords
 MSC47D09
 METIS311609
 MSC47D06
 MSC47A60
 EWI26259
 Functional calculus
 IR97140
 NWO 613.001.004
Cite this
Schwenninger, F. L. (2015). On Functional Calculus Estimates. Enschede: Universiteit Twente. https://doi.org/10.3990/1.9789036539623