Title:

Laplace transforms, nonanalytic growth bounds and C₀semigroups

In this thesis, we study a nonanalytic 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 nonanalytic abscissa of convergence, $\zeta_{1}(f)$ and the nonanalytic abscissa of absolute convergence $\kappa(f)$. These new bounds may be considered as nonanalytic 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 nonanalytic growth bound of the classical fractional integrals of $f$. The definitions of $\zeta, \zeta_{1}$ and $\kappa$ extend to the operatorvalued 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 nonanalytic 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 nonresonance condition holds. We apply our theory of nonanalytic 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 pseudospectrum is of a particular shape. Lastly, we shift our focus from nonanalytic bounds to sunreflexivity 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 $\mathbf{X}$ sunreflexive.
