The Properties And Application Of γ-radonifying Operators And R-boundedness

It has become apparent that many classical results in operator theory and harmonic analysis can be generalised from their traditional Hilbert space setting to Banach spaces, provided the notion of uniform boundedness is replaced with R -boundedness. Meanwhile,γ-radonifying operators play an important role in the study of the linear stochastic Cauchy problem.The present thesis is composed of three chapters.In the first Chapter, we introduce the developing history of operator semigroups,γ-radonifying operators and R-boundedness, and the value of the studying the related problems and the main conclusions in this paper.In the second Chapter,we introduce in details some fundamental conceptions and some important conclusions aboutγ-radonifying operators and R -boundedness,the relation betweenγ-radonifying operators and Riesz basis.In the third Chapter, first,we introduce simply C₀ -semigroups and some classical exponential stability theorems. As an application aboutγ-radonifying operators and R -boundedness, we have prived the main result in this paper as follows:Let A be the generator of a strongly continuous semigroups T = { T(t)}_t≥0 on a Banach space E .The following assertions are equivalent: (a)For all f∈γ(R,E), T * f∈γ(R ,E)(b)ω₀(A)< 0.