Graduate

One can give various rigorous definitions to the notion of "functional calculus", but a functional calculus is ultimately just a mathematically meaningful way of talking about an operator f(T), where, T is an operator and f is a function. This thesis is concerned with this concept and with one of its applications, the finding of bounds for powers of operators. It is actually this very application that has prompted the entire investigation presented here. This application is relevant to various fields, such as the numerical analysis of PDE and Markov chains. Chapter I presents various abstract approaches to the notion of "functional calculus" that are given content by three major examples: the Riesz-Dunford functional calculus, the Weyl functional calculus and the functional calculus for sectorial operators. Chapter II investigates various conditions that ensure power boundedness for operators, putting the Tadmor-Ritt condition at its center. The Riesz-Dunford calculus is instrumental for the proofs in this chapter. Chapter III investigates Pascale Vitse's use of Cauchy-Stieltjes integrals and their multipliers for obtaining bounds on powers of operators; the chapter closes with an investigation of partially power bounded operators.

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This thesis is concerned with the asymptotic behaviour of solutions of the differential-difference equation:- w'(s) = g(s)[w(s - 1) - w(s)] where g(s) is a continuous real-valued function. g(s) is assumed to have one of the following asymptotic behaviours <ul><li> algebraic</li><li>exponential algebraic</li><li>constant</li><li>zero</li><li>periodic</li></ul> Chapter 2 covers the behaviour of solutions as s→ + ∝ for g(s) ≥ 0. Chapter 3 covers the behaviour of solutions as s→ + ∝ for g(s) ≤ 0. Chapter 4 covers s→ - ∝ and g(s) ≥ 0. Chapter 5 covers s→ - ∝ and g(s) ≤ 0.

In this thesis, we study a non-analytic growth bound $\zeta(f)$ associated with an exponentially bounded measurable function $f: \mathbb{R}_{+} \to \mathbf{X},$ which measures the extent to which $f$ can be approximated by holomorphic functions. This growth bound is related to the location of the domain of holomorphy of the Laplace transform of $f$ far from the real axis. We study the properties of $\zeta(f)$ as well as two associated abscissas, namely the non-analytic abscissa of convergence, $\zeta_{1}(f)$ and the non-analytic abscissa of absolute convergence $\kappa(f)$. These new bounds may be considered as non-analytic analogues of the exponential growth bound $\omega_{0}(f)$ and the abscissas of convergence and absolute convergence of the Laplace transform of $f,$ $\operatorname{abs}(f)$ and $\operatorname{abs}(\|f\|)$. Analogues of several well known relations involving the growth bound and abscissas of convergence associated with $f$ and abscissas of holomorphy of the Laplace transform of $f$ are established. We examine the behaviour of $\zeta$ under regularisation of $f$ by convolution and obtain, in particular, estimates for the non-analytic growth bound of the classical fractional integrals of $f$. The definitions of $\zeta, \zeta_{1}$ and $\kappa$ extend to the operator-valued case also. For a $C_{0}$-semigroup $\mathbf{T}$ of operators, $\zeta(\mathbf{T})$ is closely related to the critical growth bound of $\mathbf{T}$. We obtain a characterisation of the non-analytic growth bound of $\mathbf{T}$ in terms of Fourier multiplier properties of the resolvent of the generator. Yet another characterisation of $\zeta(\mathbf{T}) $ is obtained in terms of the existence of unique mild solutions of inhomogeneous Cauchy problems for which a non-resonance condition holds. We apply our theory of non-analytic growth bounds to prove some results in which $\zeta(\mathbf{T})$ does not appear explicitly; for example, we show that all the growth bounds $\omega_{\alpha}(\mathbf{T}), \alpha >0,$ of a $C_{0}$-semigroup $\mathbf{T}$ coincide with the spectral bound $s(\mathbf{A})$, provided the pseudo-spectrum is of a particular shape. Lastly, we shift our focus from non-analytic bounds to sun-reflexivity of a Banach space with respect to $C_{0}$-semigroups. In particular, we study the relations between the existence of certain approximations of the identity on the Banach space $\xspace$ and that of $C_{0}$-semigroups on $\mathbf{X}$ which make $\mathbf{X}$ sun-reflexive.

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iii, 114 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1976