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

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Abstract
In this thesis, we study a nonanalytic 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 nonanalytic abscissa of convergence, and the nonanalytic abscissa of absolute convergence . These new bounds may be considered as nonanalytic 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 nonanalytic growth bound of the classical fractional integrals of . The definitions of and extend to the operatorvalued case also. For a semigroup of operators, is closely related to the critical growth bound of . We obtain a characterisation of the nonanalytic 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 nonresonance condition holds. We apply our theory of nonanalytic 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 pseudospectrum is of a particular shape. Lastly, we shift our focus from nonanalytic bounds to sunreflexivity 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 sunreflexive.
Item Type:  Thesis (PhD) 

Subjects:  D  G > Functional analysis 
Research Groups:  Functional Analysis Group 
ID Code:  48 
Deposited By:  Eprints Administrator 
Deposited On:  11 Mar 2004 
Last Modified:  29 May 2015 18:15 
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