The Norm Function And The Critical Point Of C₀-Semigroups

In this paper, we consider the characterization of the norm function, t→∥T(t)∥, of C₀-semigroups on Banach spaces, and the problem whether a norm-continuous (compact, differentiable) semigroup remains norm-continuous (com-pact, differentiable) at the left endpoint of its interval of norm-continuity (com-pactness, differentiability).The thesis consists of four parts. Part one states the background of subse-quent studies and the main results.Part two contains some basic knowledge involving functional analysis, realanalysis and matrix theory, and the proofs of several preliminary propositions.In part three, we studies the characterization of the norm function of C₀-semigroups. First of all, by easy reasoning a set of four basic properties is es-tablished as a necessary condition for a function to be the norm function of aC₀-semigroup. In§3.2, a growth estimate of the norm function of C₀-semigroupson ffnite-dimensional spaces is reached, which enable us to construct a functionthat satisffes the four basic properties but fails to meet the growth estimate andhence cannot be the the norm function of any C₀-semigroup on ffnite-dimensionalspace. In§3.3 we make a few assumptions in addition to the four basic proper-ties and proves their suffciency for a function to be the the norm function of aC₀-semigroups on an inffnite-dimensional Banach space.In the last part, we show by examples that a norm-continuous (compact,differentiable) semigroup may or may not remain norm-continuous (compact, d-ifferentiable) at its critical point, i.e. the left endpoint of its interval of norm- continuity (compactness, differentiability), depending on the concrete situation,and a general conclusion cannot be drawn.