Norm Continutity Of Exponentially Bounded Regularized Semigroups

Norm continutity of semigroups is a very important property, for many years, authors can not get a perfect result in terms of its infinitesimal generator or resolvent of the generator. In this paper,we ,at first,by Laplace transform and Fourier transform ,got a representation theorem for regularized semigroups in Hilbert space, under this representation theorem,we got two sufficient and necessary conditions for norm continutity of regularized semigroups.At the same time,for C₁ —regularized semigroups {S(t)}_t≥oandC₂—regularized semigroups {T(t)}_t≥o,denote by â–³(t) for S{t)C₁² — T(t)C₂²,we got a sufficient condition for norm continutity of A(t) in Hilbert space.Likewise,by generalizing representation theorems of C₀—semigroups,we got the representation theorem for regularized semigroups for x ∈ D(Aⁿ) C D(A²) in Banach space,by the help of the representation theorem, we got a characteristic condition for norm continutity of regularized semigroups.At last,we give an example which will use the norm continutity of regularized semigroups .