Functional Calculus via Transference, Double Operator Integrals and Applications

This thesis is dedicated to the study of several aspects of the theory of functional calculus. This theory considers the combination of an operator A and a function f(z) of a variable z, resulting in an operator f(A). One then attempts to study properties of the operator f(A) in terms of properties of the operator A and the function f. A classical example of a functional calculus is the calculus for diagonalizable matrices. This calculus is based on the idea that for a diagonal matrix D and a function f, f(D) should be defined by simply applying f to the entries of D. One then extends this definition to diagonalizable matrices in the obvious way. Despite its relatively straightforward construction, many nontrivial questions arise when studying this calculus. Recently, functional calculus theory has also proven useful when studying partial differential equations from a functional analytic perspective. The functional analytic viewpoint on a large class of partial differential equation leads to the study of differential operators A on infinite-dimensional spaces X. Functional calculus theory allows one to make this formal intuition precise and provides a convenient framework for studying the differential operator A as well as operators related to A. In this thesis both the functional calculus for diagonalizable matrices and the calculus for differential operators are studied. Using transference principles and double operator integrals, we link the theory of functional calculus to the area of harmonic analysis. Then we use theorems from harmonic analysis to deduce new results in functional calculus theory. We also apply functional calculus theory to a problem in the theory of the numerical approximation of the solutions of evolution equations. Since the solution of such an equation is usually hard to deal with analytically, one tries to approximate it by simpler expressions. One would then like to know whether this approximation converges, and functional calculus theory is a useful tool with which one can deal with this question. A more detailed description of the contents of this thesis is as follows. In Part I we treat some of the basic tools which will be used throughout the thesis. These include: the basics of the theory of functional calculus, some function space theory and preliminaries on vector-valued harmonic analysis, as well as the transference principles which link these three topics together. In Part II we study the functional calculus theory associated for operator semigroups. We obtain new links between harmonic analysis and functional calculus theory . In Chapter 3 this allows us to deduce properties of f(A) for a wide class of functions f. This class of functions depends heavily on geometrical aspects of the underlying space, and therefore so do the results which we obtain. By contrast, in Chapter 4 we study f(A) for a class of functions f which does not depend on geometrical properties of the underlying space. However, here the results are only valid when considering equations with initial values in so-called interpolation spaces. In Part III we study the functional calculus for diagonalizable matrices. For diagonalizable matrices A and B we determine how properties of f(B)-f(A) relate to properties of B-A. Again we relate a question in functional calculus theory to harmonic analysis, this time using the technique of double operator integration. In Part IV we apply functional calculus theory to the study of numerical approximation methods for the solutions to evolution equations. In particular, we consider a recently proposed numerical approximation method. We show that this approximation method converges for a large set of initial values and determine the corresponding rates of convergence. Then, using the theory from earlier chapters, we improve these rates of convergence for specific classes of operators. Finally, two appendices contain results which are used in earlier Chapters \ref{functional calculus for semigroup generators}. These results are of a technical nature and have been placed in appendices to improve the readability of the main text.