The Variant Spectrum Of C₀-semigroup And Spectral Properties Of Generation Operator

For a strongly continuous semigroup(T(t)t?0 with the generator A,in general,the spectral mapping theorem ?(T(t))\{0} = et?(A)may fail.We introduce its critical spectrum ?crit(T(t)),thus the spectral mapping theorem can be generalized to the form?(T(t)\{0} =et?(A)(?)?crit(T(t))\{0},t?0,(1)We called equation(1)as spectral mapping theorem of critical spectrum,the intersection of et?(A)and ?crit(T(t))may be nonempty,hence for a C0 semigroup,its critical spectrum is nonempty cannot illustrate the spectral mapping theorem is not valid.We define its variant spectrum ?v(T(t))the spectral mapping theorem can be further generalized to the following form?(T(t))\{0}=et?(A)(?)?v(T(t))\{0},t?0,(2)We called equation(2)as spectral mapping theorem of variant spectrum,the intersection of et?(A)and ?v(T(t))must be empty,so for a C0 semigroup,its variant spectrum is nonempty if and only if spectral mapping theorem is not valid.Equation(1)and equation(2)are both right for all strongly continuous semigroups,they are collectively called generalized spectral mapping theorem.