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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:249507 |
| Date | January 2002 |
| Creators | Srivastava, Sachi |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:4fa1ad4d-a117-441a-a0dd-8d7999e7cd55 : http://eprints.maths.ox.ac.uk/48/ |
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