Laplace transforms, non-analytic growth bounds and -semigroups

Srivastava, S. (2002) Laplace transforms, non-analytic growth bounds and -semigroups. PhD thesis, University of Oxford.

 Preview
PDF
654Kb

Abstract

In this thesis, we study a non-analytic growth bound associated with an exponentially bounded measurable function which measures the extent to which can be approximated by holomorphic functions. This growth bound is related to the location of the domain of holomorphy of the Laplace transform of far from the real axis. We study the properties of as well as two associated abscissas, namely the non-analytic abscissa of convergence, and the non-analytic abscissa of absolute convergence . These new bounds may be considered as non-analytic analogues of the exponential growth bound and the abscissas of convergence and absolute convergence of the Laplace transform of and . Analogues of several well known relations involving the growth bound and abscissas of convergence associated with and abscissas of holomorphy of the Laplace transform of are established. We examine the behaviour of under regularisation of by convolution and obtain, in particular, estimates for the non-analytic growth bound of the classical fractional integrals of . The definitions of and extend to the operator-valued case also. For a -semigroup of operators, is closely related to the critical growth bound of . We obtain a characterisation of the non-analytic growth bound of in terms of Fourier multiplier properties of the resolvent of the generator. Yet another characterisation of 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 does not appear explicitly; for example, we show that all the growth bounds of a -semigroup coincide with the spectral bound , 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 -semigroups. In particular, we study the relations between the existence of certain approximations of the identity on the Banach space and that of -semigroups on which make sun-reflexive.

Item Type: Thesis (PhD) D - G > Functional analysis Functional Analysis Group 48 Eprints Administrator 11 Mar 2004 20 Jul 2009 14:18

Repository Staff Only: item control page